Eigenvalue problems with indefinite weight

We consider the linear eigenvalue problem -Δu = λV(x)u, ${\displaystyle u\in D^{1,2}\_0(\Omega )}$, and its nonlinear generalization ${\displaystyle -{\Delta }_{p}u=\lambda V\left(x\right){\left|u\right|}^{p-2}u}$, ${\displaystyle u\in D^{1,p}\_0(\Omega )}$. The set Ω need not be bounded, in particular, ${\displaystyle \Omega ={ℝ}^N}$ is admitted. The weight function V may change sign and may have singular points. We show that there exists a sequence of eigenvalues ${\displaystyle {\lambda }_n\to \infty }$.