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

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 analyzed 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