A perturbed eigenvalue problem on general domains

Abstract

The perturbed eigenvalue problem $-\Delta u-{\Delta }_{p}u=\lambda V\left(x\right)u$, with $p\in (1,N)\setminus \left\{2\right\}$ and $V$ a weight function which takes nonnegative values and may have singular points, is studied in an Orlicz–Sobolev setting on general open sets from ${\mathbb{R}}^N$ with $N\ge 3$. The analysis of these problems leads to a full characterization of the set of parameters $\lambda$ for which the problem possesses nontrivial solutions as being an unbounded open interval.