Multi-bump solutions for a class of quasilinear problems involving variable exponents

Abstract

We establish the existence of multi-bump solutions for the following class of quasilinear problems $$-{\Delta }_{p(x)}u+(\lambda V(x)+Z(x))u^{p(x)-1}=f(x,u)\text{ in }{\mathbb{R}}^N,u\ge 0\text{ in }{\mathbb{R}}^N,$$ where the nonlinearity $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a continuous function having a subcritical growth and potentials $V,Z:{\mathbb{R}}^N\to \mathbb{R}$ are continuous functions verifying some hypotheses. The main tool used is the variational method.