An existence result for quasilinear elliptic equations with variable exponents

Abstract

We study the following elliptic equation involving weight and variable exponents
$$-\mbox{div}(\phi(x, |\nabla u|)\nabla u) + |u|^{p(x)-2}u = \lambda |u|^{r(x)-2}u - h(x)|u|^{s(x)-2}u $$
in $\Omega\subset\mathbb{R}^N (N\geq 3)$, with Dirichlet boundary condition, where $\phi (x,t)$ is of type $|t|^{p(x)-2}$ with continuous function $p:\overline{\Omega}\rightarrow (1,\infty)$.

Under appropriate conditions on $\phi$, by means of variational methods and a variant of the mountain pass theorem, we show that for $\lambda$ large enough there exist at least two nontrivial weak solutions for our problem. For this purpose we work on a generalized variable exponent Lebesgue-Sobolev space.