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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Nonlinear biharmonic boundary value problem

Tacksun Jung1 and Q-Heung Choi2*

Author Affiliations

1 Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea

2 Department of Mathematics Education, Inha University, Incheon, 402-751, Korea

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Boundary Value Problems 2014, 2014:30  doi:10.1186/1687-2770-2014-30

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/30


Received:28 June 2013
Accepted:8 January 2014
Published:4 February 2014

© 2014 Jung and Choi; 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

We consider the nonlinear biharmonic equation with variable coefficient and polynomial growth nonlinearity and Dirichlet boundary condition. We get two theorems. One theorem says that there exists at least one bounded solution under some condition. The other one says that there exist at least two solutions, one of which is a bounded solution and the other of which has a large norm under some condition. We obtain this result by the variational method, generalized mountain pass geometry and the critical point theory of the associated functional.

MSC: 35J20, 35J25, 35Q72.

Keywords:
biharmonic boundary value problem; polynomial growth; variational method; generalized mountain pass geometry; critical point theory; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M1">View MathML</a> condition

1 Introduction

Let Ω be a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M2">View MathML</a> with smooth boundary Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3">View MathML</a> be a square integrable function space defined on Ω. Let Δ be the elliptic operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M4">View MathML</a> be the biharmonic operator. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M5">View MathML</a>. In this paper we study the following nonlinear biharmonic equation with Dirichlet boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M6">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M7">View MathML</a> is a continuous function which changes sign in Ω.

We assume that g satisfies the following conditions:

(g1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M8">View MathML</a>,

(g2) there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M9">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M10">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M11">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M12">View MathML</a>,

(g3) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M13">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M14">View MathML</a>

(g4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M15">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M16">View MathML</a>.

We note that (g3) implies the existence of the positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M19">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M20">View MathML</a>

(1.2)

Remark 1.1 The real number ξ in the definition (g3) is not automatically nonnegative. The reason is as follows.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M24">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M24">View MathML</a>, we have two cases: one case is that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M27">View MathML</a>. The other case is that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M29">View MathML</a>. Thus ξ is not nonnegative.

Remark 1.2 We obtain the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30">View MathML</a> as follows.

By the condition (g3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M31">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M32">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M34">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M35">View MathML</a> in (1.2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M36">View MathML</a>

Thus we obtain the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30">View MathML</a>.

Remark 1.3 (i) Assumption (g4) implies that (1.1) has a trivial solution.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M38">View MathML</a>, (g2) can be dropped. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M39">View MathML</a>, it suffices that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M40">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M41">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M42">View MathML</a>.

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M44">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46">View MathML</a> is a small number, then (g1)-(g4) are satisfied.

The eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M47">View MathML</a>

has infinitely many eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49">View MathML</a>, and corresponding eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49">View MathML</a>, suitably normalized with respect to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3">View MathML</a> inner product, where each eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M48">View MathML</a> is repeated as often as its multiplicity. The eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M54">View MathML</a>

has also infinitely many eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49">View MathML</a> and corresponding eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49">View MathML</a>. We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M59">View MathML</a>, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M60">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M61">View MathML</a>.

Khanfir and Lassoued [1] showed the existence of at least one solution for the nonlinear elliptic boundary problem when g is locally Hölder continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M62">View MathML</a>. Choi and Jung [2] showed that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M63">View MathML</a>

(1.3)

has at least two nontrivial solutions when (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M66">View MathML</a>) or (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M69">View MathML</a>). The authors obtained these results by using the variational reduction method. The authors [3] also proved that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M66">View MathML</a>, (1.3) has at least three nontrivial solutions by using degree theory. Tarantello [4] also studied

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M73">View MathML</a>

(1.4)

She showed that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M75">View MathML</a>, then (1.4) has a negative solution. She obtained this result by degree theory. Micheletti and Pistoia [5] also proved that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M77">View MathML</a> then (1.4) has at least four solutions by the variational linking theorem and Leray-Schauder degree theory.

In this paper we are trying to find the weak solutions of (1.1), that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M78">View MathML</a>

where the space H is introduced in Section 2. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M79">View MathML</a>

and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M80">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M81">View MathML</a> changes sign, the open subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M83">View MathML</a> are nonempty. Now we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M84">View MathML</a>. Our main results are as follows.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85">View MathML</a>, gsatisfies (g1)-(g4) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86">View MathML</a>is bounded. Then (1.1) has at least one bounded nontrivial solution.

Theorem 1.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85">View MathML</a>, gsatisfies (g1)-(g4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86">View MathML</a>is not bounded and there exists a small<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M90">View MathML</a>. Then (1.1) has at least two solutions, (i) one of which is nontrivial and bounded, and (ii) the other of which has a large norm such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M91">View MathML</a>

The outline of Theorem 1.1 and Theorem 1.2 is as follows: In Section 2, we prove that the corresponding functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92">View MathML</a> of (1.1), which is introduced in (2.1), is continuous and Fréchet differentiable and satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M1">View MathML</a> condition. In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.2 by the variational method, the generalized mountain pass geometry and the critical point theory.

2 Palais-Smale condition

Any element u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M95">View MathML</a>

We define a subspace H of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M3">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M97">View MathML</a>

Then this is a Banach space with a norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M98">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M99">View MathML</a> and c is fixed, we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M100">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M101">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M102">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M103">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M104">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M105">View MathML</a>,

which are proved in [6].

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M106">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M107">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M108">View MathML</a> be the orthogonal projection on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M110">View MathML</a> be the orthogonal projection on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M111">View MathML</a>.

We are looking for the weak solutions of (1.1). The weak solutions of (1.1) coincide with the critical points of the associated functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M112">View MathML</a>

(2.1)

By (g1) and (g2), I is well defined. By Proposition 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113">View MathML</a> and I is Fréchet differentiable in H.

Proposition 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49">View MathML</a>, and thatgsatisfies (g1)-(g4). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92">View MathML</a>is continuous and Fréchet differentiable inHwith Fréchet derivative

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M117">View MathML</a>

(2.2)

If we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M118">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M119">View MathML</a>is continuous with respect to weak convergence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M119">View MathML</a>is compact, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M121">View MathML</a>

This implies that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M123">View MathML</a>is weakly continuous.

The proof of Proposition 2.1 is the same as that of Appendix B in [7].

Proposition 2.2 (Palais-Smale condition)

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M49">View MathML</a>, andgsatisfies (g1)-(g4). We also assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86">View MathML</a>is bounded or that there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M128">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92">View MathML</a>satisfies the Palais-Smale condition.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M130">View MathML</a> is a sequence with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M132">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M133">View MathML</a>. Then by (g2), (g3), and the Hölder inequality and the Sobolev Embedding Theorem, for large m and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M135">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M136">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30">View MathML</a> is bounded or there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M139">View MathML</a>; we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M140">View MathML</a>

(2.3)

Moreover since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M141">View MathML</a>

(2.4)

for large m and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M142">View MathML</a>, choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M143">View MathML</a> gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M144">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M145">View MathML</a> in (2.4) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M146">View MathML</a>

Thus, by (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M147">View MathML</a>

from which the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M130">View MathML</a> follows. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M130">View MathML</a> converges weakly in H. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M150">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M151">View MathML</a> compact and the weak convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M152">View MathML</a> imply the strong convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M152">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M1">View MathML</a> condition holds. □

3 Proof of Theorem 1.1

We shall show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92">View MathML</a> satisfies the generalized mountain pass geometrical assumptions.

We recall the generalized mountain pass geometry.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M156">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M157">View MathML</a> and is finite dimensional. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113">View MathML</a>, satisfies the Palais-Smale condition, and

(i) there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M159">View MathML</a> and a bounded neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M160">View MathML</a> of 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M161">View MathML</a>, and

(ii) there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M163">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M164">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M165">View MathML</a>.

Then I possesses a critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M166">View MathML</a>. Moreover b can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M167">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M168">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M169">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M170">View MathML</a> is a subspace of H such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M171">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M172">View MathML</a>

We have the following generalized mountain pass geometrical assumptions.

Lemma 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M85">View MathML</a>andgsatisfies (g1)-(g4). Then

(i) there are constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M175">View MathML</a>and a bounded neighborhood<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M160">View MathML</a>of 0 such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M177">View MathML</a>, and

(ii) there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M163">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M180">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M165">View MathML</a>, and

(iii) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M182">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M183">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M184">View MathML</a>.

Proof (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M185">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M186">View MathML</a>

Thus by (g2), (g4), and the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M187">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M188">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M175">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M192">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M193">View MathML</a>.

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M194">View MathML</a> be a ball with radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M195">View MathML</a>, e be a fixed element in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M197">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M200">View MathML</a>. We note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M201">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M202">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M204">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M205">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M206">View MathML</a>.

(iii) If we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M207">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M209">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M210">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M211">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M212">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M214">View MathML</a> great enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M215">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M217">View MathML</a>. □

Proof of Theorem 1.1 By Proposition 2.1 and Proposition 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M218">View MathML</a> and satisfies the Palais-Smale condition. By Lemma 3.1, there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M175">View MathML</a> and a bounded neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M160">View MathML</a> of 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M222">View MathML</a>, and there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M163">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M225">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M165">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M182">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M184">View MathML</a>. By the generalized mountain pass theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92">View MathML</a> has a critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M166">View MathML</a>. Moreover, b can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M232">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M233">View MathML</a>

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234">View MathML</a> a critical point of I such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M235">View MathML</a>. We claim that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M103">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M237">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M238">View MathML</a>.

In fact, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M239">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M240">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M241">View MathML</a>, b is bounded: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M242">View MathML</a>.

We can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M243">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M238">View MathML</a>. Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M245">View MathML</a>

(3.1)

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M246">View MathML</a>, from which we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234">View MathML</a> is bounded and the proof of Theorem 1.1 is complete. □

4 Proof of Theorem 1.2

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30">View MathML</a> is not bounded and there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M250">View MathML</a>. By Proposition 2.1 and Proposition 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113">View MathML</a> and satisfies the Palais-Smale condition. By Lemma 3.1 and the generalized mountain pass theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M92">View MathML</a> has a critical value b with critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M235">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M255">View MathML</a> is sufficiently small, by (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M256">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M103">View MathML</a>, from which we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M234">View MathML</a> is bounded and the proof of Theorem 1.2(i) is complete.

Next we shall prove Theorem 1.2(ii). We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M259">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M260">View MathML</a>. Let us set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M262">View MathML</a>.

Lemma 4.1Assume thatgsatisfies (g1)-(g4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M30">View MathML</a>is not bounded and there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M265">View MathML</a>. Then there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M266">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M267">View MathML</a>

(4.1)

Proof Let us choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M207">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M270">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M210">View MathML</a>. Then, by (g2), (g4), and the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M272">View MathML</a>

for small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M273">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M22">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M275">View MathML</a> great enough for each n and an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M266">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M278">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M280">View MathML</a>, so the lemma is proved. □

Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M281">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M282">View MathML</a>

Proof of Theorem 1.2(ii) We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M86">View MathML</a> is not bounded and there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M46">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M128">View MathML</a>. By Proposition 2.1 and Proposition 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M113">View MathML</a> and satisfies the Palais-Smale condition. By Lemma 4.1, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M266">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M288">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M279">View MathML</a>. We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M290">View MathML</a>. By Lemma 4.1 and the generalized mountain pass theorem, for n large enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M291">View MathML</a> is a critical value of I and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M292">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M293">View MathML</a> be a critical point of I such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M294">View MathML</a>. Then for each real number M, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M295">View MathML</a>. In fact, by contradiction, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M296">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M297">View MathML</a> imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M298">View MathML</a>

which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M299">View MathML</a> is bounded. This is absurd because of the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/30/mathml/M300">View MathML</a>. Thus we complete the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

TJ and Q-HC participated in the sequence alignment and drafted the manuscripted. Both authors read and approved the final manuscript.

Acknowledgements

This work (Choi) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2013010343).

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