ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

Title & Authors
ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION
Bin, Ge;

Abstract
This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: $\small{\{-div$$(1+\frac{|{\nabla}u|^{p(x)}}{\sqrt{1+|{\nabla}u|^{2p(x)}}}})|{\nabla}u|^{p(x)-2}{\nabla}u$$={\lambda}f(x,u)\;a.e.\;in\;{\Omega}\\u=0,\;on\;{\partial}{\Omega}}$ where $\small{{\Omega}{\subset}R^N}$ is a bounded domain with smooth boundary $\small{{\partial}{\Omega}}$, $\small{{\lambda}}$ > 0 is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter $\small{{\lambda}}$. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter $\small{{\lambda}}$ > 0. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.
Keywords
superlinear problem;p(x)-Laplacian;variational method;variable exponent Sobolev space;
Language
English
Cited by
1.
Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti–Rabinowitz Type Conditions, Journal of Dynamics and Differential Equations, 2017
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