On a bi-nonlocal p(x)-Kirchhoff equation via Krasnoselskii's genus

We study existence and multiplicity of solutions to the following bi-nonlocal p(x)-Kirchhoff equation via Krasnoselskii's genus, on the Sobolev space with variable exponent, where Ω is a bounded smooth domain of I R^N, 1 < p(x) < N, M and f are continuous functions, f is an odd function, and r > 0 is a real parameter. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence results for a class of nonlocal problems involving p(x)‐Laplacian

This paper is concerned with the existence of solutions to a class of p(x)‐Kirchhoff‐type equations with Dirichlet boundary data as follows: By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish some conditions on the existence of solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

一类有界区域上p(X)-Laplace方程解的存在性

在变指数Lebesgue空间Lp(x)(Ω)、变指数Sobolev空间W~1,p(x)(Ω)、加权变指数Lebesgue空间Lp(x)(Ω;|x~(α(x)))和加权变指数Sobolev空间W~1,p(x)(Ω;|x|~(a(x)))的基本理论体系的基础上利用山路引理得到一类p(x)-Laplace方程非平凡解的存在性.     

含临界指数的p(x)-Laplace方程解的存在性

研究了一含临界指数的p(x)-Laplace方程的Dirichlet边值问题,运用推广的集中紧致性原理,并结合山路引理得到了该问题非平凡弱解的存在性结果.     

Existence of three solutions for a Neumann problem involving the p(x)‐Laplace operator

In this paper, we study the existence of three solutions to a Neumann problem with nonstandard growth conditions. The technical approach is mainly based on three critical points theorem due to Ricceri. Copyright © 2011 John Wiley & Sons, Ltd.     

具有非标准增长条件的p(x)-Laplace方程弱解的唯一性

本文研究具有非标准增长条件的p(x) -Laplace方程,在给出弱解的先验估计的基础上,得到了弱解的唯一性.     

Second order regularity for the p(x)‐Laplace operator

In this paper, we establish second order regularity for the p(x)‐Laplace operator. This generalizes classical results known when the function p(.) is equal to some constant p > 1. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Infinitely many weak solutions for p(x)-Laplacian-like problems with Neumann condition

The paper studies the existence of infinitely many solutions for a Neumann problem involving the p(x)-Laplacian-like depending on two parameters. Our approach is based on variational methods.     

Multiple solutions to a class of inclusion problem with the p(x)-Laplacian

In this article, we obtain the existence of at least two nontrivial solutions for a nonlinear elliptic problem involving p(x)-Laplacian type operator and nonsmooth potentials. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.     

Existence result for a gradient-type elliptic system involving a pair of p(x) and q(x)-Laplacian operators

In this paper, we deal with a typical gradient elliptic system involving a pair of p(x) and q(x)-Laplacian operators. Furthermore, the system may have nonlinearities with sign-changing. Precisely, we are interested in seeking at least one weak nontrivial solution. In this way, we establish explicitly a pair of lower and upper solutions having radial forms and related to the system. By applying the theory of monotone operators, we show that the system possesses at least one non-trivial and bounded solution.     

Infinitely many solutions for polyharmonic equations of p(x)-Laplace type

We show the existence of infinitely many weak solutions to a class of quasilinear elliptic p(x)-polyharmonic Kirchhoff equations via the mountain pass principle without the (AR) condition. Furthermore, we obtain infinitely many solutions to this equation based on the genus theory, introduced by Krasnoselskii and the abstract critical point theorem (a variant of Ljusternik-Schnirelman theory) under Cerami condition.     

一类带权的p(x)-Laplace方程解的存在性

利用山路引理,嵌入定理和h(o|¨)lder不等式证明了一类带权的p(x)-Laplace方程非平凡解的存在性.     

R~N上一类p(x)-Laplacian方程的无穷多解问题

该文主要讨论了如下p(x)-Laplacian算子方程的解.其中1P-≤p(x)≤P+N.得到了上述方程在变指数Sobolev空间W~(1,p(x))(R~N)中的一列能量值趋向正无穷的解.     

Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions

This paper deals with the existence of solutions for a class of p(x)-biharmonic equations with Navier boundary conditions. The approach is based on variational methods and critical point theory. Indeed, we investigate the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of the third solution for the problem.     

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

Entropy solutions for nonlinear nonhomogeneous Neumann problems involving the generalized p(x)-Laplace operator

In this work we investigate a class ofnonlinear p(x) Laplace problems with Neumann nonhomoge-neous boundary conditions and L1 data. The techniques of entropy solutions for elliptic equationsare used to prove the existence of a solution.