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Existence of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian

Guowei Dai*, Ruyun Ma and Qiaozhen Ma

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou 730070, P.R. China

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Boundary Value Problems 2012, 2012:11  doi:10.1186/1687-2770-2012-11

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/11


Received:5 November 2011
Accepted:9 February 2012
Published:9 February 2012

© 2012 Dai 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

Using the non-smooth critical point theory we investigate the existence and multiplicity of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian.

Mathematics Subject Classification 2000: 35D05; 35J20; 35J60; 35J70.

Keywords:
p(x)-Laplacian; differential inclusion; singularity

1 Introduction

In this article, we study the existence and multiplicity of solutions for the differential inclusion problem with singular coefficients involving the p(x)-Laplacian of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M1">View MathML</a>

(1.1)

where the following conditions are satisfied:

(P) Ω is a bounded open domain in ℝN, N ≥ 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M2">View MathML</a>, 1 < p- := infΩ p(x) ≤ p+ := supΩ p(x) < +∞, λ, μ ∈ ℝ.

(A) For i = 1, 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M3">View MathML</a> for x ∈ Ω, Gi(x, u) is measurable with respect to x (for every u ∈ ℝ) and locally Lipschitz with respect to u (for a.e. x ∈ Ω), ∂Gi : Ω × ℝ → ℝ is the Clarke sub-differential of Gi and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M4">View MathML</a> for x ∈ Ω, t ∈ ℝ and ξi ∈ ∂Gi, where ci is a positive constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M5">View MathML</a>, ri(x) > qi(x) for all x ∈ Ω, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M6">View MathML</a>

(1.2)

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M7">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M8">View MathML</a>

A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M9">View MathML</a>

(1.4)

and in this case the assumption corresponding to (A) is the following

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M10">View MathML</a>, for i = 1, 2, ∂Gi : Ω × ℝ → ℝ is the Clarke sub-differential of Gi and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M11">View MathML</a> for x ∈ Ω, t ∈ ℝ and ξi Gi, where ci is a positive constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M12">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M13">View MathML</a>

(1.5)

The operator -div(|∇u|p(x)-2 u) is said to be the p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electro-magnetic field [1,2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal baro-tropic gas through a porous medium [3,4]. Another field of application of equations with variable exponent growth conditions is image processing [5]. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [6-11] for an overview of and references on this subject, and to [12-21] for the study of the p(x)-Laplacian equations and the corresponding variational problems.

Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions for Dirichlet boundary value problems with discontinuous nonlinearities has been widely investigated in recent years. Chang [22] extended the variational methods to a class of non-differentiable functionals, and directly applied the variational methods for non-differentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Later Kourogenis and Papageorgiou [23] obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the p-Laplacian with discontinuous nonlinearities. In the celebrated work [24,25], Ricceri elaborated a Ricceri-type variational principle and a three critical points theorem for the Gâteaux differentiable functional, respectively. Later, Marano and Motreanu [26,27] extended Ricceri's results to a large class of non-differentiable functionals and gave some applications to differential inclusion problems involving the p-Laplacian with discontinuous nonlinearities.

In [21], by means of the critical point theory, Fan obtain the existence and multiplicity of solutions for (1.1) under the condition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M14">View MathML</a> satisfying the Carathéodory condition for i = 1, 2, x ∈ Ω. The aim of the present article is to generalize the main results of [21] to the case of the functional of problem (1.1) is nonsmooth.

This article is organized as follows: In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function; In Section 3, we give the variational principle which is needed in the sequel; In Section 4, using the critical point theory, we prove the existence and multiplicity results for problem (1.1).

2 Preliminaries

2.1 Variable exponent Sobolev spaces

Let Ω be a bounded open subset of ℝN, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M15">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M16">View MathML</a>, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M17">View MathML</a>

On the basic properties of the space W1,p(x)(Ω) we refer to [7,28-30]. Here we display some facts which will be used later.

Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M18">View MathML</a>, define the spaces Lp(x) (Ω) and W1,p(x) (Ω) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M19">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M20">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M21">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M22">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M23">View MathML</a> the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M24">View MathML</a> in W1,p(x) (Ω) . Hereafter, we always assume that p- > 1.

Proposition 2.1. [7,31]The spaces Lp(x) (Ω) , W1,p(x) (Ω) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M25">View MathML</a>are separable and reflexive Banach spaces.

Proposition 2.2. [7,31]The conjugate space of Lp(x) (Ω) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M26">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M27">View MathML</a>. For any u Lp(x) (Ω) and v <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M29">View MathML</a>.

Proposition 2.3. [7,31]In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M30">View MathML</a>the Poincaré inequality holds, that is, there exists a positive constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M31">View MathML</a>

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M32">View MathML</a>is an equivalent norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M33">View MathML</a>.

Proposition 2.4. [7,28,29,31]Assume that the boundary of Ω possesses the cone property and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M34">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M35">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M36">View MathML</a>, then there is a compact embedding W1,p(x)(Ω) → Lq(x) (Ω).

Let us now consider the weighted variable exponent Lebesgue space.

Let a S(Ω) and a(x) > 0 for x ∈ Ω. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M37">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M38">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M39">View MathML</a> is a Banach space. The following proposition follows easily from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M40">View MathML</a>.

Proposition 2.5. (see [7,31]) Set ρ(u) = Ω a(x)|u(x)|p(x) dx. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M41">View MathML</a>, we have

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M42">View MathML</a>

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M43">View MathML</a>

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M44">View MathML</a>

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M45">View MathML</a>

(5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M46">View MathML</a>

(6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M47">View MathML</a>

Proposition 2.6. (see [21]) Assume that the boundary of Ω possesses the cone property and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M48">View MathML</a>. Suppose that a Lr(x)(Ω), a(x) > 0 for x ∈ Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M49">View MathML</a>and r- > 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M50">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M51">View MathML</a>

(2.1)

then there is a compact embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M52">View MathML</a>.

The following proposition plays an important role in the present article.

Proposition 2.7. Assume that the boundary of Ω possesses the cone property and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M53">View MathML</a>. Suppose that a Lr(x)(Ω), a(x) > 0 for x ∈ Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M54">View MathML</a>and r(x) > q(x) for all x ∈ Ω. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M55">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M56">View MathML</a>

(2.2)

then there is a compact embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M57">View MathML</a>.

Proof. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M58">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M60">View MathML</a>. Moreover, from (2.2) we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M61">View MathML</a>

Using Proposition 2.6, we see that the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M62">View MathML</a> is compact.

2.2 Generalized gradient of the locally Lipschitz function

Let (X, || · ||) be a real Banach space and X* be its topological dual. A function f : X → ℝ is called locally Lipschitz if each point u X possesses a neighborhood Ωu such that |f(u1) - f(u2)| ≤ L||u1 - u2|| for all u1, u2 ∈ Ωu, for a constant L > 0 depending on Ωu. The generalized directional derivative of f at the point u X in the direction v X is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M63">View MathML</a>

The generalized gradient of f at u X is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M64">View MathML</a>

which is a non-empty, convex and w*-compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. We say that u X is a critical point of f if 0 ∈ ∂f(u). For further details, we refer the reader to Chang [22].

We list some fundamental properties of the generalized directional derivative and gradient that will be used throughout the article.

Proposition 2.8. (see [22,32]) (1) Let j : X → ℝ be a continuously differentiable function. Then j(u) = {j'(u)}, j0(u; z) coincides with j' (u), zX and (f + j)0(u, z) = f0(u; z) + 〈j' (u), zX for all u, z X.

(2) The set-valued mapping u → ∂f(u) is upper semi-continuous in the sense that for each u0 X, ε > 0, v X, there is a δ > 0, such that for each w ∈ ∂f (u) with ||w - u0|| < δ, there is w0 ∈ ∂f (u0)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M65">View MathML</a>

(3) (Lebourg's mean value theorem) Let u and v be two points in X. Then there exists a point w in the open segment joining u and v and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M67">View MathML</a>

(4) The function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M68">View MathML</a>

exists, and is lower semi continuous; i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M69">View MathML</a>.

In the following we need the nonsmooth version of Palais-Smale condition.

Definition 2.1. We say that φ satisfies the (PS)c-condition if any sequence {un} ⊂ X such that φ(un) → c and m(un) → 0, as n → +∞, has a strongly convergent subsequence, where m(un) = inf{||u*||X* : u* ∈ ∂φ (un)}.

In what follows we write the (PS)c-condition as simply the PS-condition if it holds for every level c ∈ ℝ for the Palais-Smale condition at level c.

3 Variational principle

In this section we assume that Ω and p(x) satisfy the assumption (P). For simplicity we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M70">View MathML</a> and ||u|| = |∇u|p(x) for u X. Denote by un ⇀ u and un u the weak convergence and strong convergence of sequence {un} in X, respectively, denote by c and ci the generic positive constants, Bρ = {u X : ||u|| < ρ}, Sρ = {u X : ||u|| = ρ}.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M71">View MathML</a>

(3.1)

where ai and Gi (i = 1, 2) are as in (A).

Define the integral functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M72">View MathML</a>

(3.2)

We write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M73">View MathML</a>

then it is easy to see that J C1(X, ℝ) and φ = J - Ψ.

Below we give several propositions that will be used later.

Proposition 3.1. (see [19]) The functional J : X → ℝ is convex. The mapping J' : X X* is a strictly monotone, bounded homeomorphism, and is of (S+) type, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M74">View MathML</a>

Proposition 3.2. Ψ is weakly-strongly continuous, i.e., un u implies Ψ(un) → Ψ(u).

Proof. Define ϒ1 = ∫Ω G1(x, u) dx and ϒ2 = ∫Ω G2(x, u) dx. In order to prove Ψ is weakly-strongly continuous, we only need to prove ϒ1 and ϒ2 are weakly-strongly continuous. Since the proofs of ϒ1 and ϒ2 are identical, we will just prove ϒ1.

We assume un u in X. Then by Proposition 2.8.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M75">View MathML</a>

where ξn ∈ ∂G1(,τn(x)) for some τn(x) in the open segment joining u and un. From Chang [22] we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M76">View MathML</a>. So using Proposition 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M77">View MathML</a>

Proposition 3.3. Assume (A) holds and F satisfies the following condition:

(B) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M78">View MathML</a>for a.e.x ∈ Ω, all u X and ξ1 ∈ ∂G1, ξ2 ∈ ∂G2, where θ is a constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M80">View MathML</a>.

Then φ satisfies the nonsmooth (PS) condition on X.

Proof. Let {un} be a nonsmooth (PS) sequence, then by (B) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M81">View MathML</a>

and consequently {un} is bounded.

Thus by passing to a subsequence if necessary, we may assume that un u in X as n → ∞. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M82">View MathML</a>

with εn ↓ 0, where ξin(x) ∈ ∂Gi(x, un) for a.e. x ∈ Ω, i = 1, 2. From Chang [22] or Theorem 1.3.10 of [33], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M83">View MathML</a>. Since X is embedded compactly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M84">View MathML</a>, we have that un u as n → ∞ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M85">View MathML</a>. So using Proposition 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M86">View MathML</a>

Therefore we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M87">View MathML</a>. But we know that J' is a mapping of type (S+). Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M88">View MathML</a>

Remark 3.1. Note that our condition (1.2) is stronger than (1.2) of [21]. Because Ψ' is weakly-strongly continuous in [21], to verify that φ satisfies (PS) condition on X, it is enough to verify that any (PS) sequence is bounded. However, in this paper we do not know whether ξ(u) is weakly-strongly continuous, where ξ(u) ∈ ⇀Ψ. Therefore, it will be very useful to consider this problem.

Below we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M89">View MathML</a>

We shall use the following conditions.

(B1) ∃ c0 > 0 such that G2(x, t) ≥ - c0 for x ∈ Ω and t ∈ ℝ.

(B2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M90">View MathML</a> and M > 0 such that 0 < G2(x, u) ≤ θ u, ξ2〉 for x∈ Ω, u X and |u| M, ξ2 ∈ ⇀G2.

Corollary 3.1. Assume (P), (A) and (A1) hold. Then φ satisfies nonsmooth (PS) condition on X provided either one of the following conditions is satisfied.

(1). λ ∈ ℝ and μ = 0.

(2). λ ∈ ℝ, μ = 0 and (B1) holds.

(3). λ ∈ ℝ, μ ∈ ℝ and (B2) holds.

Proof. In case (1) or (2), we have, for x ∈ Ω and t ∈ ℝ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M91">View MathML</a>

which shows that the condition (B) with θ = 0 is satisfied.

In case (3), noting that (B2) and (A) imply (B1), by the conclusion (1) and (2) we know φ satisfies (PS) condition if μ ≤ 0. Below assume μ > 0. The conditions (B2) and (A) imply that, for x ∈ Ω and u X,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M92">View MathML</a>

so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M93">View MathML</a>

which shows (B) holds. The proof is complete. ■

As X is a separable and reflexive Banach space, there exist (see [[34], Section 17]) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M94">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M95">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M96">View MathML</a>

For k = 1, 2, . . . , denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M97">View MathML</a>

(3.3)

Proposition 3.5. [35]Assume that Ψ : X → ℝ is weakly-strongly continuous and Ψ (0) = 0. Let γ > 0 be given. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M98">View MathML</a>

Then βk → 0 as k → ∞.

Proposition 3.6. (Nonsmooth Mountain pass theorem, see [23,33]) If X is a reflexive Banach space, φ : X → ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)c-condition, and for some r > 0 and e1 X with ||e1|| > r, max{φ(0), φ(e1)} ≤·inf{φ(u) : ||u|| = r}. Then φ has a nontrivial critical u X such that the critical value c = φ(u) is characterized by the following minimax principle

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M99">View MathML</a>

where Γ = {γ C([0, 1], X) : γ(0) = 0, γ(1) = e1}.

Proposition 3.7. (Nonsmooth Fountain theorem, see [36]) Assume (F1) X is a Banach space, φ : X → ℝ be an invariant locally Lipschitz functional, the subspaces Xk, Yk and Zk are defined by (3.3).

If, for every k ∈ ℕ, there exist ρk > rk > 0 such that

(F2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M100">View MathML</a>

(F3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M101">View MathML</a>

(F4) φ satisfies the nonsmooth (PS)c condition for every c > 0, then φ has an unbounded sequence of critical values.

Proposition 3.8. (Nonsmooth dual Fountain theorem, see [37]) Assume (F1) is satisfied and there is a k0 > 0 such that, for each k k0, there exists ρk > γk > 0 such that

(D1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M102">View MathML</a>

(D2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M103">View MathML</a>

(D3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M104">View MathML</a>

(D4) φ satisfies the nonsmooth <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M105">View MathML</a>condition for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M106">View MathML</a>, then φ has a sequence of negative critical values converging to 0.

Remark 3.2. We say φ that satisfies the nonsmooth <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M107">View MathML</a> condition at level c ∈ ℝ (with respect to (Yn)) if any sequence {un} ⊂ X such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M108">View MathML</a>

contains a subsequence converging to a critical point of φ.

4 Existence and multiplicity of solutions

In this section, using the critical point theory, we give the existence and multiplicity results for problem (1.1). We shall use the following assumptions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M110">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M111">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M112">View MathML</a>

(S) For i = 1, 2, Gi(x, -t) = Gi(x, t), ∀x ∈ Ω, ∀t ∈ ℝ.

Remark 4.1.

(1) It follows from (A), (A2) and (O2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M113">View MathML</a>

(2)It follows from (A) and (B2) that (see [33, p. 298])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M114">View MathML</a>

The following is the main result of this article.

Theorem 4.1. Assume (P), (A), (A1) hold.

(1) If (B1) holds, then for every λ ∈ ℝ and μ ≤ 0, problem (1.1) has a solution which is a minimizer of the corresponding functional φ.

(2) If (B1), (A2), (O1), (O2) hold, then for every λ > 0 and μ ≤ 0, problem (1.1) has a nontrivial solution v1 such that v1 is a minimizer of φ and φ(v1) < 0.

(3) If (A2), (B2), (O2) hold, then for every μ > 0, there exists λ0(μ) > 0 such that when |λ| ≤ λ0(μ), problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0.

(4) If (A2), (B2), (O1), (O2) holds, then for every μ > 0, there exists λ0(μ) > 0 such that when 0 < λ λ0(μ), problem (1.1) has two nontrivial solutions u1 and v1 such that φ(u1) > 0 and φ(v1) < 0.

(5) If (A2), (B2), (O1), (O2) and (S) holds, then for every μ > 0 and λ ∈ ℝ, problem (1.1) has a sequence of solutions uk} such that φuk) → ∞ as k → ∞.

(6) If (A2), (B2), (O1), (O2) and (S) holds, then for every λ > 0 and μ ∈ ℝ, problem (1.1) has a sequence of solutions vk} such that φvk) < 0 and φvk) → 0 as k → ∞.

Proof. We will use c, c' and ci as a generic positive constant. By Corollary 3.1, under the assumptions of Theorem 4.1, φ satisfies nonsmooth (PS) condition. We write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M115">View MathML</a>

then Ψ = Ψ1 + Ψ2, φ(u) = J(u) - Ψ (u) = J(u) - Ψ1(u) - Ψ2(u). Firstly, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M116">View MathML</a> to denote its extension to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M117">View MathML</a>, where i = 1, 2. From (A) and Theorem 1.3.10 of [33] (or Chang [22]), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M118">View MathML</a>(u) is locally Lipschitz on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M120">View MathML</a> for a.e. x ∈ Ω and i = 1, 2. In view of Proposition 2.4 and Theorem 2.2 of [22], we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M121">View MathML</a> is also locally Lipschitz, and ∂Ψ1(u) ⊆ λ Ω a1(x) ∂G1(x, u) dx, ∂Ψ2(u) ⊆ μ Ω a2(x) ∂G1(x, u) dx, (see [38]), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M122">View MathML</a> stands for the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M123">View MathML</a> to X for i = 1, 2. Therefore, φ is a locally Lipschitz functional on X.

(1) Let λ ∈ ℝ and μ ≤ 0. By (A),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M124">View MathML</a>

By (B1), Ψ2(u) ≤ - μc0 Ω a2(x) dx = c5. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M125">View MathML</a>. By (A1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M126">View MathML</a>, so φ is coercive, that is, φ(u) → ∞ as ||u|| → ∞. Thus φ has a minimize which is a solution of (1.1).

(2) Let λ > 0, μ ≤ 0 and the assumptions of (2) hold. By the above conclusion (1), φ has a minimize v1. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M127">View MathML</a> such that 0 ≤ v0(x) ≤ min{δ1, δ2}, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M129">View MathML</a>. By (O1) and (O2) we have, for t ∈ (0, 1) small enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M130">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M131">View MathML</a>, we can find t0 ∈ (0, 1) such that φ(t0v0) < 0, and this shows φ(v1) = infuX φ(u) < 0. So v1 ≠ 0 because φ(0) = 0. The conclusion (2) is proved.

(3) Let μ > 0 and the assumptions of (3) hold. By Remark 4.1.(1), for sufficiently small ||u||

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M132">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M134">View MathML</a>, there exists γ > 0 and α > 0 such that J(u) - Ψ2(u) ≥ α for u Sγ. We can find λ0(μ) > 0 such that when |λ| ≤ λ0(μ), Ψ1(u) ≤ α/2 for u Sγ. So when |λ| ≤ λ0(μ), φ(u) ≥ α/2 > 0 for u Sγ. By Remark 4.1.(2), noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M135">View MathML</a>, we can find a u0 X such that ||u0|| > γ and φ(u0) < 0. By Proposition 3.6 problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0.

(4) Let μ > 0 and the assumptions of (4) hold. By the conclusion (3), we know that, there exists λ0(μ) > 0 such that when 0 < λ λ0(μ), problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0. Let γ and α be as in the proof of (3), that is, φ(u) ≥ α/2 > 0 for u Sγ. By (O1), (O2) and the proof of (2), there exists w X such that ||w|| < γ and φ(w) < 0. It is clear that there is v1 Bγ, a minimizer of φ on Bγ. Thus v1 is a nontrivial solution of (1.1) and φ(v1) < 0.

(5) Let μ > 0, λ ∈ ℝ and the assumptions of (5) hold. By (S), we can use the nonsmooth version Fountain theorem with the antipodal action of ℤ2 to prove (5) (see Proposition 3.7). Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M136">View MathML</a>

Let βk(γ) be as in Proposition 3.5. By Proposition 3.5, for each positive integer n, there exists a positive integer k0(n) such that βk(n) ≤ 1 for all k k0(n). We may assume k0(n) < k0(n + 1) for each n. We define {γk : k = 1, 2, . . . , } by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M137">View MathML</a>

Note that γk → ∞ as k → ∞. Then for u Zk with ||u|| = γk we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M138">View MathML</a>

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M139">View MathML</a>

i.e., the condition (F2) of Proposition 3.7 is satisfied.

By (A), (A1), (B2) and Remark 4.1.(2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M140">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M141">View MathML</a> and all norms on a finite dimensional vector space are equivalent each other, we can see that, for each Yk, φ(u) → - ∞ as u Yk and ||u|| → ∞. Thus for each k there exists ρk > γk such that φ(u) < 0 for u Yk Sρk, so the condition (F3) of Proposition 3.7 is satisfied. As was noted earlier, φ satisfies nonsmooth (PS) condition. By Proposition 3.7 the conclusion (5) is true.

(6) Let λ > 0, μ ∈ ℝ and the assumptions of (5) hold. Let us verify the conditions of the Nonsmooth dual Fountain theorem (see Proposition 3.8). By (S), φ is invariant on the antipodal action of ℤ2. For Ψ(u) = ∫Ω F(x, u)dx = Ψ1(u)+ Ψ2(u) let βk(1) be as in Proposition 3.5, that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M142">View MathML</a>

By Proposition 3.5, there exists a positive integer k0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M143">View MathML</a> for all k k0. Setting ρk = 1, then for k k0 and u Zk S1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M144">View MathML</a>

which shows that the condition (D1) of Proposition 3.8 is satisfied.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M145">View MathML</a> is the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M146">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M147">View MathML</a>, we may choose {Yk : k = 1, 2, . . . , }, a sequence of finite dimensional vector subspaces of X defined by (3.5), such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M148">View MathML</a> for all k. For each Yk, because all norms on Yk are equivalent each other, there is ε ∈ (0, 1) such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M150">View MathML</a> By (O1) and (O2), for u Yk Bε we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M151">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M152">View MathML</a> there exists γk ∈ (0, ε) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M153">View MathML</a>

thus the condition (D2) of Proposition 3.8 is satisfied.

Because Yk Zk ≠ ∅ and γk < ρk, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M154">View MathML</a>

On the other hand, for any u Zk with ||u|| ≤ 1 = ρk, we have φ(u) = J(u) - Ψ(u) ≥ -Ψ(u) ≥ k(1). Noting that βk → 0 as k → ∞, we obtain dk → 0, i.e., (D3) of Proposition 3.8 is satisfied.

Finally let us prove that φ satisfies nonsmooth <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M155">View MathML</a> condition for every c R. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M156">View MathML</a>. Similar to the process of verifying the (PS) condition in the proof of Proposition 3.3, we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M157">View MathML</a> in X. Let us prove 0 ∈ ∂φ(u) below. Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M158">View MathML</a>

Using Proposition 2.8.4, Going to limit in the right side of above equation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M159">View MathML</a>

so m(u) ≡ 0, i.e., 0 ∈ ∂φ(u), this shows that φ satisfies the nonsmooth <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M160">View MathML</a> condition for every c ∈ ℝ. So all conditions of Proposition 3.8 are satisfied and the conclusion (6) follows from Proposition 3.8. The proof of Theorem 4.1 is complete.    ■

Remark 4.2

Theorem 4.1 includes several important special cases. In particular, in the case of the problem (1.4), i.e., in the case that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/11/mathml/M161">View MathML</a>

all conditions of Theorem 4.1 are satisfied provided (P), (A*), (A1), and (A2) hold.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. RM participated in the design of the study. All authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030, 10971087), 1107RJZA223 and the Fundamental Research Funds for the Gansu Universities.

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