Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains

Abstract

In this paper, we consider the semilinear elliptic problem − Δ u − μ u | x | 2 − λ u = K ( x ) | u | 2 * − 2 u in Ω, u = 0 on ∂ Ω, where Ω is a smooth bounded domain in R N , N ⩾ 4 , 0 < μ < ( N − 2 ) 2 4 , 2 ∗ : = 2 N N − 2 is the critical Sobolev exponent, K ( x ) is a continuous function. When Ω and K ( x ) are invariant under a group of orthogonal transformations, we prove the existence of nodal and positive solutions for 0 < λ < λ 1 , where λ 1 is the first Dirichlet eigenvalue of − Δ − μ | x | 2 on Ω.