Positive solutions to a semilinear elliptic equation with a Sobolev–Hardy term

Abstract

In this paper, we study the existence of positive solutions to the following semilinear elliptic equation with a Sobolev–Hardy term (0.1) { − Δ u − λ u = u 2 ♯ − 1 | y | x ∈ Ω , u > 0 , x ∈ Ω , u ∈ H 0 1 ( Ω ) , where Ω is a bounded domain with smooth boundary in R N ( N ≥ 3 ) , x = ( y , z ) ∈ Ω ⊂ R k × R N − k = R N , 2 ≤ k < N , 2 ♯ ≔ 2 ( N − 1 ) N − 2 is the corresponding critical exponent and 0 < λ < λ 1 where λ 1 is the first eigenvalue of − Δ in H 0 1 ( Ω ) . When N ≥ 4 , we prove that problem (0.1) has at least one positive solution by using the mountain-pass lemma and a global compactness result. The case N = 3 is quite different and we deal with this case by using the method in Jannelli (1999) [20] to prove the existence result. Moreover, we obtain the nonexistence result of (0.1) in a star shaped domain. Our main results extend a recent result of Castorina et al. (2009) [10] where λ = 0 and Ω = R N .