Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces

Abstract

In this paper we consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential $V$ on a bounded domain in $\mathbb{R}^N$ ($N\geq 3$) with a smooth boundary. We establish three main results with various assumptions. The first one asserts that any $\lambda\gt 0$ is an eigenvalue of our problem. The second theorem states the existence of a constant $\lambda_{*}\gt 0$ such that any $\lambda\in(0,\lambda_{*}]$ is an eigenvalue, while the third theorem claims the existence of a constant $\lambda^{*}\gt 0$ such that every $\lambda\in[\lambda^{*}, \infty)$ is an eigenvalue of the problem.