Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Minimax Theory and its Applications 02 (2017), No. 1, 079--097Copyright 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 vincentiu.radulescu@imar.ro Ionela-Loredana Stancut Dept. of Mathematics, University of Craiova, 200585 Craiova, Romania stancutloredana@yahoo.com [Abstract-pdf] 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 [ Fulltext-pdf  (192  KB)] for subscribers only.