On an elliptic problem with critical exponent and Hardy potential

Abstract

In this paper, we study the following elliptic problem with critical exponent and a Hardy potential: − Δ u − μ | x | 2 u = λ u + | u | 2 ⁎ − 2 u , u ∈ H 0 1 ( Ω ) , where Ω is a smooth open bounded domain in R N ( N ⩾ 3 ) which contains the origin and 2 ⁎ is the critical Sobolev exponent. We show that, if N ⩾ 5 and μ ∈ ( 0 , ( N − 2 2 ) 2 − ( N + 2 N ) 2 ) , this problem has a ground state solution for each fixed λ > 0 . Moreover, we give energy estimates from below and bounds on the number of nodal domains for these ground state solutions. If N ⩾ 7 and μ ∈ ( 0 , ( N − 2 2 ) 2 − 4 ) , this problem has infinitely many sign-changing solutions for each fixed λ > 0 .