Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential

Abstract

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ≥ 5 , 0 ≤ s < 2 , 2 ∗ ( s ) = 2 ( N − s ) N − 2 . We prove the existence of nontrivial solutions for the singular critical problem − Δ u − μ u | x | 2 = | u | 2 ∗ ( s ) − 2 | x | s u + λ u with Dirichlet boundary condition on Ω for all λ > 0 and 0 ≤ μ < ( N − 2 2 ) 2 − ( N + 2 N ) 2 .