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Minimax Theory and its Applications 02 (2017), No. 1, 079--097
Copyright Heldermann Verlag 2017

Perturbation Effects for a Singular Elliptic Problem with Lack of Compactness and Critical Exponent

Vicentiu D. Radulescu
Dept. of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Ionela-Loredana Stancut
Dept. of Mathematics, University of Craiova, 200585 Craiova, Romania


We study the existence of multiple weak entire solutions of the nonlinear elliptic equation $$ -\Delta u=V(x)|x|^{\alpha}|u|^{\frac{2(\alpha+2)}{N-2}}u +\lambda g(x)\ \ \ \text{in}\ \mathbb{R}^{N}\ (N\geq 3), $$ where $V(x)$ is a positive potential, $\alpha\in(-2,0)$, $\lambda$ is a positive parameter, and $g$ belongs to an appropriate weighted Sobolev space. We are concerned with the perturbation effects of the potential $g$ and we establish the existence of some $\lambda_{*}>0$ such that our problem has two solutions for all $\lambda\in(0,\lambda_{*})$, hence for small perturbations of the right-hand side. A first solution is a local minimum near the origin, while the second solution is obtained as a mountain pass. The proof combines the Ekeland variational principle, the mountain pass theorem without the Palais-Smale condition, and a weighted version of the Brezis-Lieb lemma.

Keywords: Singular elliptic equation, Caffarelli-Kohn-Nirenberg inequality, perturbation, critical point, weighted Sobolev space.

MSC: 35B20, 35B33, 35J20, 58E05

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