On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a potential $V$. The problem is analyzed in the context of Orlicz-Sobolev spaces. Connected with this problem we also study the optimization problem for the particular eigenvalue given by the infimum of the Rayleigh quotient associated to the problem with respect to the potential $V$ when $V$ lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.