Singular elliptic problems with lack of compactness
Marius Ghergu, Vicentiu Radulescu
TL;DR
We study a nonlinear singular elliptic equation with weight in $\mathbb{R}^N$, $-\operatorname{div}(|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\lambda g(x)$, where $p$ is the Caffarelli-Kohn-Nirenberg critical exponent associated to $a$, $b$, and $N$, and $g$ belongs to a weighted Sobolev space $H^1_a(\mathbb{R}^N)$. Under natural assumptions on the positive potential $K$, there exists $\lambda_0>0$ such that for all $\lambda\in(0,\lambda_0)$ the problem has at least two distinct weak solutions; the proof relies on Ekeland's variational principle and the Mountain Pass theorem without the Palais-Smale condition, aided by a weighted Brezis-Lieb lemma. The analysis introduces energy functionals $J_0$ and $I$, a PS-compatibility lemma ensuring that a weak limit of a $(PS)_c$ sequence is a solution, and a weighted Brezis-Lieb lemma; it also establishes a PS-dichotomy involving the CK-N optimal constant $S(a,b)$ and shows the existence of a local minimax level $c_{0,\lambda}$ yielding a ground-state solution $u_0$, and a Mountain Pass level $c_g$ yielding a second critical point $u_1$. If $g\ge0$, a maximum principle implies both solutions can be taken positive, giving multiplicity for small $\lambda$.
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.