Existence of Solutions for a Robin Problem Involving the p(x)-Laplace Operator

Abstract

In this article we study the nonlinear Robin boundary-value problem $-{\mathrm{\Delta }}_{p(x)}u=f(x,u)\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }$, $|\nabla u{|}^{p\left(x\right)-\mathrm{2}}(\partial u/\partial \nu )+\beta (x){\left|u\right|}^{p(x)-\mathrm{2}}u=\mathrm{0}$ on $\partial \mathrm{\Omega }$. Using the variational method, under appropriate assumptions on $f$, we obtain results on existence and multiplicity of solutions.