Singular elliptic problems with lack of compactness

2005/02/04 by Marius Ghergu, Ghergu, Marius, Vicentiu Radulescu +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.math/0502096

arxiv created 2005/02/04 · arxiv updated 2009/12/01

Abstract

We consider the following nonlinear singular elliptic equation -div (|x|^-2a∇ u)=K(x)|x|^-bp|u|^p-2u+\la g(x) 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∈(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.

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