Singular elliptic problems with lack of compactness

By Marius Ghergu, Vicentiu D. Rădulescu. 2005

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We consider the following nonlinear singular elliptic equation $$-\mbox{div}\,(|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\lambda{g}(x)\quad\mbox{in}\,\,\mathbb{R}^N,$$ 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 λ0>0 such that the above problem has at least two distinct solutions provided that λ∈(0,λ0). 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/15f2dc9de13b07669427b08db9c3d75e378b1ee3