On a class of nonhomogenous quasilinear problems in Orlicz-Sobolev spaces

Abstract

We study the nonlinear boundary value problem $-div ((a_1(|\nabla u(x)|)+a_2(|\nabla u(x)|))\nabla u(x))=\lambda |u|^{q(x)-2}u-\mu |u|^{\alpha(x)-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $\lambda$, $\mu$ are positive real numbers, $q$ and $\alpha$ are continuous functions and $a_1$, $a_2$ are two mappings such that $a_{1}(|t|)t$, $a_{2}(|t|)t$, are increasing homeomorphisms from $\mathbb{R}$ to $\mathbb{R}$. The problem is analysed in the context of Orlicz-Soboev spaces. First we show the existence of infinitely many weak solutions for any $\lambda,\mu \gt 0$. Second we prove that for any $\mu \gt 0$, there exists $\lambda_$ sufficiently small, and $\lambda^$ large enough such that for any $\lambda \in (0,\lambda_)\cup(\lambda^,\infty)$, the above nonhomogeneous quasilinear problem has a non-trivial weak solution.