Perturbation Effects for a Singular Elliptic Problem with Lack of Compactness and Critical Exponent
Keywords:
Singular elliptic equation, Caffarelli-Kohn-Nirenberg inequality, perturbation, criti cal point, weighted Sobolev space.Abstract
We study the existence of multiple weak entire solutions of the nonlinear elliptic equation −∆u=V(x)|x|α|u|2(α+2) N−2 u+λg(x) in RN (N ≥3), where V(x) is a positive potential, α ∈ (−2,0), λ 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 λ∗ > 0 such that our problem has two solutions for all λ ∈ (0,λ∗), 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.
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