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Existence and multiplicity of solutions for a class of sublinear Schrödinger-Maxwell equations

Ying Lv

Author Affiliations

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

Boundary Value Problems 2013, 2013:177  doi:10.1186/1687-2770-2013-177


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/177


Received:24 September 2012
Accepted:5 July 2013
Published:30 July 2013

© 2013 Lv; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper I consider a class of sublinear Schrödinger-Maxwell equations, and new results about the existence and multiplicity of solutions are obtained by using the minimizing theorem and the dual fountain theorem respectively.

Keywords:
Schrödinger-Maxwell equations; sublinear; minimizing theorem; dual fountain theorem

1 Introduction and main result

Consider the following semilinear Schrödinger-Maxwell equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M1">View MathML</a>

(1)

Such a system, also known as the nonlinear Schrödinger-Poisson system, arises in an interesting physical context. Indeed, according to a classical model, the interaction of a charge particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger and the Maxwell equations (we refer to [1,2] for more details on the physical aspects and on the qualitative properties of the solutions). In particular, if we are looking for electrostatic-type solutions, we just have to solve (1).

In recent years, system (1), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M2">View MathML</a> or being radially symmetric, has been widely studied under various conditions on f; see, for example, [3-11]. Since (1) is set on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3">View MathML</a>, it is well known that the Sobolev embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M5">View MathML</a>) is not compact, and then it is usually difficult to prove that a minimizing sequence or a sequence that satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M6">View MathML</a> condition, briefly a Palais-Smale sequence, is strongly convergent if we seek solutions of (1) by variational methods. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7">View MathML</a> is radial (for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M8">View MathML</a>), we can avoid the lack of compactness of Sobolev embedding by looking for solutions of (1) in the subspace of radial functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M9">View MathML</a>, which is usually denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M10">View MathML</a>, since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M11">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M12">View MathML</a>) is compact. Specially, Ruiz [11] dealt with (1) under the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M14">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M15">View MathML</a>) and got some general existence, nonexistence and multiplicity results.

Moreover, in [12] the authors considered system (1) with periodic potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7">View MathML</a>, and the existence of infinitely many geometrically distinct solutions was proved by the nonlinear superposition principle established in [13].

There are also some papers treating the case with nonradial potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7">View MathML</a>. More precisely, Wang and Zhou [14] got the existence and nonexistence results of (1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M18">View MathML</a> is asymptotically linear at infinity. Chen and Tang [15] proved that (1) has infinitely many high energy solutions under the condition that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M19">View MathML</a> is superlinear at infinity in u by the fountain theorem. Soon after, Li, Su and Wei [16] improved their results.

Up to now, there have been few works concerning the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M7">View MathML</a> is nonradial potential and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M19">View MathML</a> is sublinear at infinity in u. Very recently, Sun [17] treated the above case based on the variant fountain theorem established in Zou [18].

Theorem 1.1[17]

Assume that the following conditions hold:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M23">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M24">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M25">View MathML</a>is a constant. For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M27">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M28">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M29">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M31">View MathML</a>is a positive function such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M33">View MathML</a>.

Then problem (1) has infinitely many nontrivial solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M34">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M35">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>.

In the present paper, based on the dual fountain theorem, we can prove the same result under a more generic condition, which generalizes the result in [17]. Our first result can be stated as follows.

Theorem 1.2Assume thatVsatisfies

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M23">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M39">View MathML</a>;

andfsatisfies the following conditions.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40">View MathML</a>) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M42">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M43">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M44">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M46">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47">View MathML</a>) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M49">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M50">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M51">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M53">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54">View MathML</a>) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M55">View MathML</a>, there exist a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M56">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M57">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M58">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M60">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M61">View MathML</a>) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M63">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M64">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M65">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M66">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M67">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M69">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M70">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M71">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M73">View MathML</a>.

Then problem (1) has infinitely many nontrivial solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M34">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M75">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>.

By Theorem 1.2, we obtain the following corollary.

Corollary 1.3Assume thatLsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>) andWsatisfies

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M78">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M29">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M33">View MathML</a>is a constant and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M82">View MathML</a>is a function such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M84">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M66">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M68">View MathML</a>.

Then problem (1) has infinitely many nontrivial solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M34">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M88">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>.

Remark 1.4 In Theorem 1.2, infinitely many solutions for problem (1) are obtained under the symmetry condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M90">View MathML</a>) by using the dual fountain theorem. As a special case of Theorem 1.2, Corollary 1.3 generalizes and improves Theorem 1.1. To show this, it suffices to compare (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M28">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M78">View MathML</a>). Firstly, it is clear that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>) is really weaker than (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22">View MathML</a>). Secondly, in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M28">View MathML</a>) a is assumed to be positive, while in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M78">View MathML</a>) we assume that a is indefinite.

Moreover, under all the conditions of Theorem 1.2 except (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M70">View MathML</a>) we obtain an existence result.

Theorem 1.5Assume thatLsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>) andWsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M61">View MathML</a>). Then problem (1) possesses a nontrivial solution.

Remark 1.6 In Theorem 1.5 we obtain the existence of solutions for problem (1) under the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M105">View MathML</a> is indefinite and without any coercive assumptions respect to V such as (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M22">View MathML</a>). There are functions V and f which satisfy Theorem 1.5, but do not satisfy the corresponding results in [2-16]. For example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M107">View MathML</a>

(2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M108">View MathML</a>

(3)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M109">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M110">View MathML</a> is indefinite. Denoting by π the area of the unit ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M112">View MathML</a>

(4)

which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M113">View MathML</a>. So, (2) satisfies our results, but does not satisfy the results in [3-17].

2 Preliminary results

In order to establish our results via critical point theory, we firstly describe some properties of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M9">View MathML</a>, on which the variational functional associated with problem (1) is defined. Define the function space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M115">View MathML</a>

equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M116">View MathML</a>

and the function space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M117">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M118">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M119">View MathML</a>

equipped with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M120">View MathML</a>

and the corresponding norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M121">View MathML</a>

Note that the following embeddings

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M122">View MathML</a>

are continuous, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M123">View MathML</a> is the critical exponent for the Sobolev embeddings in dimension 3. Therefore, there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M125">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M126">View MathML</a>

(5)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M128">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M129">View MathML</a>) denotes the Banach spaces of a function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3">View MathML</a> with values in R under the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M131">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M132">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M84">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a> is a Banach space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M136">View MathML</a>

Lemma 2.1Suppose that assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>) holds. Then the embedding ofEin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>is compact, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a>is positive for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>.

Proof For any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M142">View MathML</a>, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M143">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M144">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M145">View MathML</a>. We claim that K is precompact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>. In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M149">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M150">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M151">View MathML</a>, applying the Hölder inequality for r such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M152">View MathML</a> and the first inequality in (5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M153">View MathML</a>

(6)

Besides, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M154">View MathML</a> is compactly embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M155">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M156">View MathML</a>, there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M157">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M145">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M159">View MathML</a>

(7)

Now it follows from (6) and (7) that K is precompact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>. Obviously, we have E is compact embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a> is positive for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>. □

Lemma 2.2Assume that assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54">View MathML</a>) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M169">View MathML</a>inE. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M170">View MathML</a>

in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M171">View MathML</a>.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M169">View MathML</a> in E. Then, by Lemma 2.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M173">View MathML</a>

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M174">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a> is positive for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>. Passing to a subsequence if necessary, it can be assumed that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M178">View MathML</a>

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M179">View MathML</a>

(8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M180">View MathML</a>

(9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M181">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M182">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>, so by (9) we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M184">View MathML</a> is also a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>. Therefore, by the completeness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M187">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M188">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M135">View MathML</a>. Now we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M190">View MathML</a>

(10)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M191">View MathML</a> and almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>. If not, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M194">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M195">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M196">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M197">View MathML</a>. Then there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M199">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M200">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M201">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M202">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M203">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M204">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M202">View MathML</a>. Therefore, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M207">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M209">View MathML</a>

which contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M84">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>. Now we have proved (10). It follows from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47">View MathML</a>) that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M26">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M214">View MathML</a>

(11)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M53">View MathML</a>. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40">View MathML</a>), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M41">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M219">View MathML</a>

(12)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M46">View MathML</a>, which together with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54">View MathML</a>) shows there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M223">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M224">View MathML</a>

(13)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M226">View MathML</a>. Combining (11) and (13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M227">View MathML</a>

(14)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M73">View MathML</a>. Hence, by (10) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M230">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M233">View MathML</a>

(15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M231">View MathML</a>. By the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M235">View MathML</a>

(16)

Similarly, we can prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M236">View MathML</a>

(17)

also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M237">View MathML</a>

(18)

It follows from (15), (16), (17) and (18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M238">View MathML</a>

which together with Lebesgue’s convergence theorem shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M239">View MathML</a>

(19)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M240">View MathML</a>. Now we have proved the lemma. □

In the proof of Theorem 1.2, the following lemma is needed.

Lemma 2.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M241">View MathML</a>is an open set. Then, for any closed set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M242">View MathML</a>, there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M243">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M244">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M246">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M247">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M248">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M249">View MathML</a>.

Proof Letting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M250">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M252">View MathML</a>. For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148">View MathML</a>, defining α and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M254">View MathML</a> as follows,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M255">View MathML</a>

one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M258">View MathML</a>. Denoting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M259">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M260">View MathML</a>

it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M262">View MathML</a>. Lastly, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M263">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M264">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M246">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M244">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M268">View MathML</a>. Moreover, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M254">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M271">View MathML</a>. □

Since E is a Hilbert space, then there exists a basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M272">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M273">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M274">View MathML</a>. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M276">View MathML</a>, now we show the following lemma, which will be used in the proof of Theorem 1.2.

Lemma 2.4Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M32">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M279">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>.

Proof It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M281">View MathML</a>, so there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M282">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M283">View MathML</a>

(20)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M285">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M286">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M287">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M288">View MathML</a>

(21)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M289">View MathML</a> is bounded, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M291">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>. Now, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M293">View MathML</a> is a basis of E, it follows that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M294">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M295">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M296">View MathML</a>, which shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M297">View MathML</a>. By Lemma 2.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M298">View MathML</a>

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M174">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a>, which together with (20) and (21) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M302">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a>. □

We obtain the existence of a solution for problem (1) by using the following standard minimizing argument.

Lemma 2.5[19]

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M305">View MathML</a>satisfying the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M6">View MathML</a>condition. If Φ is bounded from below,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M307">View MathML</a>

is a critical value of Φ.

In order to prove the multiplicity of solutions, we will use the dual fountain theorem. Firstly, we introduce the definition of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308">View MathML</a> condition.

Definition 2.6 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M310">View MathML</a>. The function Φ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308">View MathML</a> condition if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M312">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M313">View MathML</a>

contains a subsequence converging to a critical point of Φ.

Now we show the following dual fountain theorem.

Lemma 2.7[20]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M314">View MathML</a>and for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M205">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M316">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M317">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M318">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M319">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>.

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M321">View MathML</a>satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308">View MathML</a>condition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M323">View MathML</a>, then Φ has a sequence of critical points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M324">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M325">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>.

3 Proof of theorems

Define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M327">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M328">View MathML</a>

(22)

It is easy to know that I exhibits a strong indefiniteness, namely it is unbounded both from below and from above on an infinitely dimensional subspace. This indefiniteness can be removed using the reduction method described in [1], by which we are led to study a variable functional that does not present such a strong indefinite nature.

Now we recall this method. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a>, consider the linear functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M330">View MathML</a> defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M331">View MathML</a>

By the Hölder inequality and using the second inequality in (5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M332">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M333">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M334">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M335">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M336">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M337">View MathML</a> is bilinear, bounded and coercive. Hence, the Lax-Milgram theorem implies that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a>, there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M339">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M340">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M341">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M342">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M341">View MathML</a>. Using integration by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M344">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M341">View MathML</a>, therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M346">View MathML</a>

(23)

in a weak sense. We can write an integral expression for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M347">View MathML</a> in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M348">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M349">View MathML</a> (see [21], Theorem 1); by density it can be extended for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a> (see Lemma 2.1 of [22]). Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M351">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M352">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a>.

It follows from (23) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M354">View MathML</a>

(24)

and by the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M355">View MathML</a>

and it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M356">View MathML</a>

(25)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M357">View MathML</a>

(26)

So, we can consider the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M358">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M359">View MathML</a>. By (24), the reduced functional takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M360">View MathML</a>

(27)

By (12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M361">View MathML</a>

(28)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M46">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M365">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M367">View MathML</a>, the space of continuous function u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M3">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M369">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M370">View MathML</a>. Therefore there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M371">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M372">View MathML</a>

(29)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M373">View MathML</a>. Hence, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M374">View MathML</a>

which together with (26) shows that Φ is well defined. Furthermore, it is well known that Φ is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M375">View MathML</a> functional with derivative given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M376">View MathML</a>

It can be proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M377">View MathML</a> is a solution of problem (1) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a> is a critical point of the functional Φ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M379">View MathML</a>; see, for instance, [1].

Lemma 3.1Under conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54">View MathML</a>), Φ satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308">View MathML</a>condition.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M385">View MathML</a> is a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M386">View MathML</a>

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M387">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M388">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M389">View MathML</a>.

Firstly, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M390">View MathML</a> is bounded. By (14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M391">View MathML</a>

(30)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>, which together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M394">View MathML</a> implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M395">View MathML</a>

(31)

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M396">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M397">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M398">View MathML</a> is bounded.

By the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M390">View MathML</a> is bounded in E, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M401">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M402">View MathML</a>

(32)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M403">View MathML</a>

in E as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>. It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M405">View MathML</a>

(33)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M406">View MathML</a>

(34)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>. On the other hand, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>), (32) and Lemma 2.2, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M409">View MathML</a>

(35)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M411">View MathML</a>

(36)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>. Summing up (33) and (36), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M413">View MathML</a>

(37)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>. By the Hölder inequality and (25), one gets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M415">View MathML</a>

Then by Lebesgue’s convergence theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M416">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>, which together with (34) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M418">View MathML</a>

(38)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>. By Lemma 2.2 and (32), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M420">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>. Moreover, an easy computation shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M422">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M423">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M404">View MathML</a>. Φ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M308">View MathML</a> condition. □

Remark 3.2 Under conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M37">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M40">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M47">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M54">View MathML</a>), Φ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M6">View MathML</a> condition. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M431">View MathML</a> is a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M432">View MathML</a> is bounded and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M433">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M240">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M387">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M436">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M231">View MathML</a>. The rest of the proof is the same as that of Lemma 3.1.

Proof of Theorem 1.2 For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M438">View MathML</a>, we take k disjoint open sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M439">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M440">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M442">View MathML</a>, there exist a closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M443">View MathML</a> and an open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M444">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M445">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M446">View MathML</a>

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M444">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M448">View MathML</a>), by Lemma 2.3 there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M449">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M450">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M451">View MathML</a>. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M452">View MathML</a>, can be extended to be a basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M272">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M273">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M274">View MathML</a>. Now we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M456">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M457">View MathML</a>.

By Lemma 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M309">View MathML</a> satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M459">View MathML</a> condition and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M460">View MathML</a>. Hence, to prove Theorem 1.2, we should just show that Φ has the geometric property (i), (ii) and (iii) in Lemma 2.7.

(i) By Lemma 2.4

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M461">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M83">View MathML</a>. In view of (30) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M465">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M466">View MathML</a>

(39)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M467">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M468">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M469">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M470">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>. Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M472">View MathML</a>

(40)

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M473">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M474">View MathML</a>. Now we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M475">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M476">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>. When k is large enough, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M474">View MathML</a>, which together with (40) shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M480">View MathML</a>

(ii) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M482">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M483">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M484">View MathML</a>

(41)

and also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M485">View MathML</a>

(42)

Since all the norms of a finite dimensional space are equivalent, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M486">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M487">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481">View MathML</a>. By (30), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M489">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M490">View MathML</a>

(43)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M492">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M493">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M494">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M448">View MathML</a>, which together with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M497">View MathML</a>) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M498">View MathML</a>

(44)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M499">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M448">View MathML</a>. Combining (24), (41), (42), (43) and (44), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M501">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M481">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M503">View MathML</a>, when ε and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M504">View MathML</a> are both small enough. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M505">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M506">View MathML</a> small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M507">View MathML</a>

(iii) By (40), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M508">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M509">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M510">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M511">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M512">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M514">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>.

Hence, by Lemma 2.7, we obtain that problem (1) has infinitely many solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M516">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M517">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M36">View MathML</a>. □

Proof of Theorem 1.5 Similar to (31), there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M519">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M397">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M521">View MathML</a>

(45)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M127">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M523">View MathML</a>, it follows from (45) that the functional Φ is bounded from below. By Lemma 2.5 and Remark 3.2, Φ possesses a critical point u satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M524">View MathML</a>

It remains to show that u is nontrivial. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M148">View MathML</a>, there exist an open set G and a closed set H such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M526">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M527">View MathML</a>

By Lemma 2.3, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M243">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M530">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M531">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M532">View MathML</a>. Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M533">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M534">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>, which together with (28) shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M536">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>. Therefore, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M538">View MathML</a>

(46)

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M539">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M540">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M45">View MathML</a>, which together with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M497">View MathML</a>) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M543">View MathML</a>

(47)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M66">View MathML</a>. It follows from (24), (46), (47) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M545">View MathML</a>

when ε and λ are both small enough. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M546">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M547">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/177/mathml/M548">View MathML</a> is a nontrivial solution of problem (1). □

Competing interests

The author declares that she has no competing interests.

Acknowledgements

The author is highly grateful for the referees’ careful reading and comments on this paper. This work is partially supported by the National Natural Science Foundation of China (No. 11071198) and Southwest University Doctoral Fund Project (SWU112107).

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