Multiple positive solutions for a nonlinear elliptic equation in weighted Sobolev space

Abstract

In this paper, we consider the problem (${\cal P}_{\lambda}$) in the setting of a weighted Sobolev space $W^{1, p}(\Omega, \omega)$, where $\omega$ is a weight function defined on the unbounded domain $\Omega$. The study is based on the variational methods and critical point theory. We show the existence of at least two nonnegative solutions, one with negative energy, the other one with energy which changes sign at a certain value of the positive parameter $\lambda$.