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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

Existence of nontrivial solutions for p-Kirchhoff type equations

Chunhan Liu*, Jianguo Wang and Qingling Gao

Author Affiliations

Department of Mathematics, Qilu Normal University, Jinan, 250013, P.R. China

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Boundary Value Problems 2013, 2013:279  doi:10.1186/1687-2770-2013-279

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/279


Received:13 July 2013
Accepted:29 November 2013
Published:30 December 2013

© 2013 Liu et al.; licensee Springer.

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

Abstract

In this paper, the linking theorem and the mountain pass theorem are used to show the existence of nontrivial solutions for the p-Kirchhoff equations without assuming Ambrosetti-Rabinowitz type growth conditions, nontrivial solutions are obtained.

MSC: 35J60, 35J25.

Keywords:
linking theorem; mountain pass theorem; nontrivial solutions

1 Introduction

In this paper, we consider the nonlocal elliptic problem of the p-Kirchhoff type given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M2">View MathML</a> is a bounded domain, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M3">View MathML</a> is the p-Laplacian with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M4">View MathML</a>.

Recently, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M5">View MathML</a>

(2)

began to attract the attention of several researchers only after Lion [1] had proposed an abstract framework for this problem. Perera and Zhang [2] obtained a nontrivial solution of (2) by using the Yang index and critical group. They revisited (2) via invariant sets of decent flow and obtained the existence of a positive solution, a negative, and a sign-changing solutions in [3].

The study of Kirchhoff-type equations has been extended to the following case involving the p-Laplacian:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M6">View MathML</a>

for details see [4-6]. One of the authors has done some related work on this field. Liu [7] gave infinite solutions to the following equation via the fountain theorem and the dual fountain theorem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M7">View MathML</a>

However, to the best of our knowledge, there have been few papers dealing with equation (1) using the linking theorem and the mountain pass theorem. This paper will make some contribution to this research field.

It is well known (see [8]) that the eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M8">View MathML</a>

has the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M9">View MathML</a>, which is simple, and has an associated eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M10">View MathML</a>. It is also known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M11">View MathML</a> is an isolated point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M12">View MathML</a>, the spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M13">View MathML</a>, which contains at least an eigenvalue sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M15">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M16">View MathML</a>

be a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M17">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M18">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M19">View MathML</a> be the one-dimensional eigenspace associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M11">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M21">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M22">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M23">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M24">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M25">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M26">View MathML</a>, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M27">View MathML</a>, the second eigenvalue of −△ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M28">View MathML</a>.

In this paper, the weak solutions of (1) are the critical points of the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M29">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M31">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M32">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M34">View MathML</a>

In this paper we use the following notation: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M35">View MathML</a> denotes the Lebesgue space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M36">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M37">View MathML</a> denotes the Lebesgue measure of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M2">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M39">View MathML</a> is the dual pairing of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41">View MathML</a>; → (resp. ⇀) denotes strong (resp. weak) convergence. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M42">View MathML</a> denote positive constants (possibly different).

Definition 1[9]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M43">View MathML</a>, we say that Φ satisfies the Cerami condition at the level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M44">View MathML</a> if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M45">View MathML</a>, along with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M46">View MathML</a>

possesses a convergent subsequence; Φ satisfies the (C) condition if Φ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M47">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M44">View MathML</a>.

Definition 2[9]

A subset A of E is link (with respect to Φ ) to B of E if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M49">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M50">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M51">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M52">View MathML</a>.

Theorem 1[9] (Linking theorem)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M53">View MathML</a>be a Banach space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M54">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M55">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M56">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M57">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M58">View MathML</a>

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M43">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M60">View MathML</a>

If Φ satisfies the (PS) condition with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M61">View MathML</a>

thencis a critical value of Φ.

Remark 1 If Φ satisfies the (C) condition, then Theorem 1 still holds.

Theorem 2[10] (Mountain pass theorem)

LetXbe a real Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M43">View MathML</a>satisfy the (C) condition. Suppose, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M64">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M66">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M67">View MathML</a>

Then Φ has a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M68">View MathML</a>characterized by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M69">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M70">View MathML</a>

2 Main results

In this section, we give our main theorem. Near the origin, we make the following assumptions.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M71">View MathML</a> is a continuous function satisfying the following conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72">View MathML</a>) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M73">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M74">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M75">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M76">View MathML</a>) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M77">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M78">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M81">View MathML</a>.

Caratheodory function f satisfies:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>) For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M83">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M85">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M89">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M90">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M91">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M94">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M96">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M97">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M100">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M104">View MathML</a>.

The main results of this paper are the following.

Theorem 3Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M76">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102">View MathML</a>) hold, then problem (1) has at least one nontrivial weak solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41">View MathML</a>.

Theorem 4Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M76">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99">View MathML</a>hold, then problem (1) has at least one nontrivial weak solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41">View MathML</a>.

3 Proofs of theorems

First, we give several lemmas.

Lemma 1[7]

Under assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>), any bounded sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M119">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M120">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M40">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M122">View MathML</a>has a convergent subsequence.

Lemma 2Under assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>), the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M125">View MathML</a>satisfies the (C) condition.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M119">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M127">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M128">View MathML</a>

(3)

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M129">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41">View MathML</a>. For this purpose, we can suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M131">View MathML</a>. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M133">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M134">View MathML</a>

(4)

For large n, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M135">View MathML</a>, (3) and (4) imply that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M136">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M137">View MathML</a>

This is a contradiction. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M138">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M139">View MathML</a>. By Lemma 1, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M138">View MathML</a> has a convergent subsequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M41">View MathML</a>. □

Proof of Theorem 3 We obtain from assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M96">View MathML</a>) that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M145">View MathML</a> small, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M146">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M147">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M148">View MathML</a>, using the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M149">View MathML</a> and the Sobolev inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M150">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M151">View MathML</a>

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M152">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M153">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M157">View MathML</a>.

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M158">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M161">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M162">View MathML</a>. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102">View MathML</a>), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M164">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M165">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M166">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M167">View MathML</a>

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M90">View MathML</a>), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M64">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M172">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M173">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M174">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M175">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M176">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M177">View MathML</a> large enough, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M178">View MathML</a>.

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M55">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M180">View MathML</a>

By Lemmas 1 and 2, Φ satisfies the (C) condition. Then the conclusion follows from Theorem 1 and Remark 1. □

Remark 2 (i) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M181">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M187">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M188">View MathML</a>

Hence (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>) is much weaker than Ambrosetti-Rabinowitz type growth conditions.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M190">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M191">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M92">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M193">View MathML</a>

Example 1 Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M194">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M196">View MathML</a>. Then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197">View MathML</a> satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102">View MathML</a>) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M200">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M201">View MathML</a>, we can use odd expansion to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197">View MathML</a>.

Example 2 Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M203">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M204">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M207">View MathML</a>. Then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197">View MathML</a> satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M102">View MathML</a>) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M200">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M201">View MathML</a>, we can use odd expansion to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197">View MathML</a>.

Proof of Theorem 4 We obtain from assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M72">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99">View MathML</a> that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M145">View MathML</a> small, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M218">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M219">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M18">View MathML</a>, using the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M221">View MathML</a> and the Sobolev inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M222">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M223">View MathML</a>

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M64">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M225">View MathML</a>.

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M230">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M231">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M232">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M233">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M234">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M235">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M236">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M237">View MathML</a>

Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M231">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M240">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M241">View MathML</a>

Hence there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M243">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M244">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M245">View MathML</a>

Summing up Lemma 1 and Lemma 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M125">View MathML</a> satisfies all the conditions of Theorem 2, then the conclusion follows from Theorem 2. □

Remark 3 The result of Theorem 1.1 in [11] corresponds to our results for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M248">View MathML</a> replaces (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>). It is easy to see that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>) is much weaker than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M248">View MathML</a>, hence the results of Theorems 3 and 4 extend the results of [11].

Example 3 Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M252">View MathML</a>

where p is odd, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M253">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M254">View MathML</a>

where p is even, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M253">View MathML</a>.

Then similar to [12], it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M197">View MathML</a> satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M82">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M86">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M99">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/279/mathml/M200">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CL and JW obtained the results in a joint research. All the authors read and approved the final manuscript.

Acknowledgements

This work was supported by the National Nature Science Foundation of China (10971179) and the Project of Shandong Province Higher Educational Science and Technology Program (J09LA55, J12LI53).

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