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http://dx.doi.org/10.14317/jami.2014.137

INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH

Zhou, Chenxing (College of Mathematics, Changchun Normal University)
Liang, Sihua (College of Mathematics, Changchun Normal University)

Publication Information

Journal of applied mathematics & informatics / v.32, no.1_2, 2014 , pp. 137-152 More about this Journal

Abstract

In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$ . We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$ , where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

Keywords

p(x)-Laplacian; Generalized Lebesgue-Sobolev spaces; Symmetric mountain-pass lemma; Concentration-compactness principle;

Citations & Related Records

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