Singular elliptic problems with lack of compactness

Abstract

We consider the following nonlinear singular elliptic equation $$-{div} (|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\la g(x) \quad{in} \RR^N,$$ where $g$ belongs to an appropriate weighted Sobolev space, and $p$ denotes the Caffarelli-Kohn-Nirenberg critical exponent associated to $a$, $b$, and $N$. Under some natural assumptions on the positive potential $K(x)$ we establish the existence of some $\la\_0>0$ such that the above problem has at least two distinct solutions provided that $\la\in(0,\la\_0)$. The proof relies on Ekeland's Variational Principle and on the Mountain Pass Theorem without the Palais-Smale condition, combined with a weighted variant of the Brezis-Lieb Lemma.