Infinitely many solutions for a class of superlinear problems involving variable exponents

Abstract

We are concerned with the following $p(x)$-Laplacian equations in $\mathbb{R}^N$ $$ -\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u= f(x,u)\ \ {\rm in }\; \mathbb{R}^N. $$ Based on a direct sum decomposition of a space, we investigate the multiplicity of solutions for the considered problem. The potential $V$ is allowed to be no coerciveness, and the primitive of the nonlinearity $f$ is of super-$p^+$ growth near infinity in $u$ and allowed to be sign-changing. Our assumptions are suitable and different from those studied previously.