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Multiple solutions for a p-Laplacian elliptic problem

Jing Zeng1 and Shuting Cai2*

Author Affiliations

1 School of Mathematics and Computer Sciences, Fujian Normal University, Fuzhou, 350007, P.R. China

2 Department of Mathematics and Physics, Fujian Jiangxia University, Fuzhou, 350108, P.R. China

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Boundary Value Problems 2014, 2014:124  doi:10.1186/1687-2770-2014-124

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/124


Received:17 April 2014
Accepted:6 May 2014
Published:20 May 2014

© 2014 Zeng and Cai; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

We consider the following p-Laplacian elliptic equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M1">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M2">View MathML</a>. For certain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M3">View MathML</a>, we are interested in the functional on a group invariant subspace, and we obtain the existence of infinitely many radial solutions and non-radial solutions of the equation, which extends the result of (Bartsch and Willem in J. Funct. Anal. 117:447-460, 1993) to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M4">View MathML</a>.

Keywords:
p-Laplacian; infinitely many radial solutions and non-radial solutions; group invariant

1 Introduction

The interesting equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M5">View MathML</a>

(1.1)

originates from different problems in physics and mathematical physics. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M6">View MathML</a>, problem (1.1) is interpreted as a stationary state of the reaction-diffusion Klein-Gordon equation in chemical dynamics, and Schrödinger equations in finding certain solitary waves.

In the 1980s, people searched for the spherically symmetric solutions of the autonomous equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M7">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M8">View MathML</a> is continuous and odd in u). Berestycki-Lions [1,2] advocated it for the first time; they obtained the existence of infinitely many radial solutions of the autonomous equation. Then Struwe [3] got similar results. Gidas et al.[4] further demonstrated that any positive solution of the equation with some properties must be radial.

Then Bartsch-Willem [5] found an unbounded sequence of non-radial solutions of (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M6">View MathML</a>, under the assumption that b and f satisfy certain growth conditions and f is odd in u.

In recent years, the existence and structure of solutions for the p-Laplacian equation has found considerable interest, and different approaches have been developed. Bartsch-Liu [6] studied the p-Laplacian problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M11">View MathML</a>

(1.2)

on a bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M12">View MathML</a> with smooth boundary Ω, provided that the nonlinearity f is superlinear and subcritical. They proved (1.2) has a pair of a subsolution and a supersolution. In [7] they studied problem (1.2) on a bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M13">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M14">View MathML</a> arbitrary, and proved a nodal solution provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M15">View MathML</a> is subcritical and superlinear. Infinitely many nodal solutions are obtained if, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M16">View MathML</a>.

Furthermore Jiu-Su [8] applied Morse theory to study the existence of nontrivial solutions of p-Laplacian type Dirichlet boundary value problems. Agarwal-Perera [9] obtained two positive solutions of singular discrete p-Laplacian problems using variational methods. Chabrowski-Fu [10] studied the p-Laplacian problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M17">View MathML</a>

on a bounded domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18">View MathML</a> with Dirichlet boundary condition, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M19">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M21">View MathML</a> are positive constants). They applied the mountain pass theorem to prove the existence of solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M22">View MathML</a> for the equation in the superlinear and sublinear cases.

For (1.1), Drábek-Pohozaev [11] proved the existence of multiple positive solutions of quasilinear problems (1.1) of second order by using the fibering method. They considered solutions both in the bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M13">View MathML</a> and in the whole space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18">View MathML</a>. Moreover, De Nápoli-Mariani [12] introduced a notion of uniformly convex functional that generalizes the notion of uniformly convex norm. They proved the existence of at least one solution of (1.1), and the existence of infinitely many solutions under further assumptions.

In the present paper, we aim to find the existence of infinitely many radial and non-radial solutions of problem (1.1), and extend the result of [5] to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M4">View MathML</a>.

A direct extension to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M26">View MathML</a> is faced with serious difficulties. First the energy functional associated to (1.1) is defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M1">View MathML</a>, which is not a Hilbert space for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M26">View MathML</a>. Another difficulty is the lack of a powerful regularity theory. For the Laplace operator there exists a sequence of Banach spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M29">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M31">View MathML</a>. But the imbedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M32">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M33">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M34">View MathML</a>) is not compact.

We study the functional on a group invariant subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M35">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M36">View MathML</a> is the group of orthogonal linear transformations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18">View MathML</a>), then we apply the principle of symmetric criticality [[13], Theorem 5.4] and the fountain theorem to obtain the existence of multiple solutions.

2 The main results and preliminaries

This paper is devoted to the study of infinitely many radial and non-radial solutions for a p-Laplacian equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M38">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M39">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M34">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M41">View MathML</a> is bounded from below by a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M42">View MathML</a>. The growth condition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M3">View MathML</a> will be given in the following.

The corresponding functional is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M44">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M46">View MathML</a>.

We require the following assumptions on the nonlinearity f:

(f1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M47">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M48">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M49">View MathML</a>, uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M50">View MathML</a>.

(f2) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M51">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M54">View MathML</a>.

(f3) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M34">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M56">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M57">View MathML</a>, such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M60">View MathML</a>.

The main result of this paper is as follows.

Theorem 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M61">View MathML</a>, the assumptions (f1)-(f3) hold, andfis odd inu, then problem (1.1) possesses infinitely many radial solutions.

Theorem 2.2Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M62">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M63">View MathML</a>, if the assumptions (f1)-(f3) hold andfis odd inu, then for problem (1.1) there exist infinitely many non-radial solutions.

Remark 2.3 The assumptions (f1)-(f3) are from [5]. Bartsch-Willem [5] considered the existence of non-radial solutions for the Euclidean scalar field equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M64">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M65">View MathML</a>).

(f2) means that the nonlinearity f is superlinear, and (f3) means that f is subcritical. These two conditions enable us to use a variational approach for the study of (1.1).

Condition (f2) corresponds to the standard superlinearity condition of Ambrosetti-Rabinowitz in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M66">View MathML</a>. In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M6">View MathML</a> without the assumption (f3), the above theorems may not be true. It can be seen from Pohozaev’s identity for p-Laplacian equations that (1.1) has only a trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M68">View MathML</a>.

Remark 2.4 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M69">View MathML</a> we fail to define the action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M70">View MathML</a> in the proof of Theorem 2.2.

We shall use the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M71">View MathML</a>

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M72">View MathML</a> is the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M73">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M74">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M75">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M76">View MathML</a> the usual norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M77">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M78">View MathML</a> be the group of orthogonal linear transformations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M80">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M81">View MathML</a>.

Throughout this paper, we will use C and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M82">View MathML</a> to represent various positive constants.

Now, we recall some definitions for the action of a topological group and the fountain theorem.

Definition 2.5 ([[14], Definition 1.27])

The action of a topological group G on a normed space Z is a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M83">View MathML</a>, such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M85">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M86">View MathML</a>

The action is isometric if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M87">View MathML</a>.

The set of invariant points is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M88">View MathML</a>. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M89">View MathML</a> is G-invariant if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M90">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M91">View MathML</a>. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M92">View MathML</a> is G-invariant if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M93">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M91">View MathML</a>.

A map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M95">View MathML</a> is G-equivariant if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M96">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M91">View MathML</a>.

Definition 2.6 ([[15], p.99])

Suppose Z is a G-Banach space, that is, there is a G isometric action on Z. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M98">View MathML</a> be the set of G-invariant subsets of Z, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M99">View MathML</a> be the class of G-equivariant mappings of Z.

Definition 2.7 ([14])

Let Z be a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M101">View MathML</a>. The functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M102">View MathML</a> condition if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M103">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M104">View MathML</a>

has a convergent subsequence.

Theorem 2.8 (Fountain theorem [[14], Theorem 3.6])

The compact groupGacts isometrically on the Banach space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M105">View MathML</a>, the spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M106">View MathML</a>are invariant and there exists a finite-dimensional spaceVsuch that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M108">View MathML</a>. The action ofGonVis admissible.

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M109">View MathML</a>be anG-invariant functional. If for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M110">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M111">View MathML</a>such that

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M112">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M113">View MathML</a>.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M114">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M115">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M116">View MathML</a>.

(A3) Isatisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M102">View MathML</a>condition, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M118">View MathML</a>.

ThenIpossesses an unbounded sequence of critical values<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M119">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M119">View MathML</a>can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M121">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M123">View MathML</a>.

In fact, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M124">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M125">View MathML</a>, then there exists a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M126">View MathML</a>.

3 Proof of theorems

Definition 3.1 ([[14], Definition A.3])

On the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M127">View MathML</a>, we define the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M128">View MathML</a>

On the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M129">View MathML</a>, we define the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M130">View MathML</a>

Lemma 3.2 ([[14], Theorem A.4])

Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M47">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M133">View MathML</a>, then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M135">View MathML</a>, the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M136">View MathML</a>

is continuous.

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M137">View MathML</a>be the mapping given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M138">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139">View MathML</a>is bounded and continuous.

Proof By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M141">View MathML</a>

(3.1)

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139">View MathML</a> is bounded. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M143">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>, by (3.1), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M145">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139">View MathML</a> is continuous. □

Lemma 3.4Suppose the nonlinearityfsatisfies (f1)-(f3), then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M147">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M148">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M149">View MathML</a>, here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M150">View MathML</a>is compact. In addition, each critical point ofJis a weak solution of problem (1.1).

Proof By Lemma 3.3, we only need to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M150">View MathML</a> is continuous. By Hölder inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M152">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M154">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M155">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M157">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M158">View MathML</a>. It follows from (f3) and Lemma 3.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M159">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M160">View MathML</a>

(3.2)

Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M161">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M163">View MathML</a> is compact, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M155">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M165">View MathML</a>. By Lemma 3.2 and (3.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M150">View MathML</a> is compact. □

Lemma 3.5 ([[16], Lemma 2.1])

There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M167">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M168">View MathML</a>, such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M170">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M171">View MathML</a>

Lemma 3.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139">View MathML</a>be defined in Lemma 3.3. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M173">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M175">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M143">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M173">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M182">View MathML</a>, by Lemma 3.5,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M183">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M143">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M186">View MathML</a>, by Lemma 3.5 and the Hölder inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M187">View MathML</a>

 □

Lemma 3.7Assume thatfsatisfies (f1)-(f3). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M188">View MathML</a>be a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M189">View MathML</a>

(3.3)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180">View MathML</a>has a subsequence which converges to a critical point of the functionalJ.

Proof First we show that each sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M188">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M193">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M194">View MathML</a>, is bounded. By (f2) and (f3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M195">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M51">View MathML</a> in the assumption (f2), so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>.

Since E is reflexive, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a> is reflexive, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M180">View MathML</a> has a weakly convergent subsequence. Going if necessary to a subsequence, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M201">View MathML</a>. By Lemma 3.4 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M202">View MathML</a>, and by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M204">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M205">View MathML</a> converges. So we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M206">View MathML</a>. Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M207">View MathML</a>

(3.4)

By Lemma 3.6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M208">View MathML</a>.

Next we want to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M209">View MathML</a> is a critical point of J, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M210">View MathML</a>. By Lemma 3.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M211">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M212">View MathML</a>

By (3.3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M209">View MathML</a> is a critical point of J. □

Now we give the proof of Theorems 2.1 and 2.2 by applying the fountain theorem and the principle of symmetric criticality. First we recall some properties of Banach space.

According to the results in [17], there exists a Schauder basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M214">View MathML</a> for E. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M215">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M216">View MathML</a> is a Schauder basis for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a> is reflexive, there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M219">View MathML</a>, which are characterized by the relations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M220">View MathML</a>, forming a basis for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M221">View MathML</a>.

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M222">View MathML</a>

and define a group action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M223">View MathML</a>.

Lemma 3.8If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M56">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M225">View MathML</a>

Proof It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M226">View MathML</a>, so we assume for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M228">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M115">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M230">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M231">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M233">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M235">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M144">View MathML</a>. Since the imbedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M163">View MathML</a> is compact, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M238">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M165">View MathML</a>. Thus we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M240">View MathML</a>. □

Proof of Theorem 2.1 Note that J is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M241">View MathML</a>-invariant, by the principle of symmetric criticality [[13], Theorem 5.4], any critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M242">View MathML</a> is a solution of problem (1.1). J is invariant with respect to the action <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M243">View MathML</a>.

Now we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M242">View MathML</a> satisfies the assumptions of the fountain theorem.

By the assumptions (f1) and (f3), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M245">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M246">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M247">View MathML</a> is the lower bound of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M248">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M249">View MathML</a>, by Lemma 3.8, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M252">View MathML</a>, and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M115">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M254">View MathML</a>

This proves (A2).

Now we want to show that the condition (A1) is satisfied. By integrating, we obtain from (f2) and (f3) that, there exist two constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M255">View MathML</a>, such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M257">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M258">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M259">View MathML</a> is finite dimensional, all norms are equivalent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M260">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M262">View MathML</a> imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M263">View MathML</a>

So there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M111">View MathML</a> such that (A1) is satisfied.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M102">View MathML</a> condition is proved above. By Theorem 2.8, we find, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M124">View MathML</a>, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M267">View MathML</a>

are critical values of the functional J. So we can get an unbounded sequence of solutions of (1.1), and the solutions are radial. □

Proof of Theorem 2.2 In this proof, we will show that it suffices to find the critical points of J restricted to a subspace of invariant functions. The proof is similar to Theorem 1.31 in [14].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M268">View MathML</a> be a fixed integer different from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M269">View MathML</a>. The action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M270">View MathML</a> on E is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M271">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M18">View MathML</a> is compatible with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M70">View MathML</a>, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M274">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M275">View MathML</a>) is compact (or see [18] for details).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M276">View MathML</a> be the involution defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M277">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M278">View MathML</a>

The action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M279">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M280">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M281">View MathML</a>

It is clear that 0 is the only radial function on the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M282">View MathML</a>

Moreover, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M283">View MathML</a> is compact. As in the proof of Theorem 2.1, we obtain a sequence of non-radial solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/124/mathml/M284">View MathML</a> of (1.1). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally. Both authors read and approved the final manuscript.

Acknowledgements

The first author is supported by the project of ‘Min-Hong Kong cooperation postdoctoral training’ funded by Fujian Provincial Civil Service Bureau and the Nonlinear Analysis Innovation Team (IRTL1206) funded by Fujian Normal University. The second author is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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