Opuscula Math. 36, no. 1 (), 81-101
http://dx.doi.org/10.7494/OpMath.2016.36.1.81
Opuscula Mathematica

# 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.
Keywords: anisotropic Orlicz-Sobolev space, potential, critical point, weak solution, eigenvalue.
Mathematics Subject Classification: 35D30, 35J60, 58E05.