On an elliptic problem with critical exponent and Hardy potential

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Abstract

In this paper, we study the following elliptic problem with critical exponent and a Hardy potential:Δuμ|x|2u=λu+|u|22u,uH01(Ω), where Ω is a smooth open bounded domain in RN (N3) which contains the origin and 2 is the critical Sobolev exponent. We show that, if N5 and μ(0,(N22)2(N+2N)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 N7 and μ(0,(N22)24), this problem has infinitely many sign-changing solutions for each fixed λ>0.

MSC

35J20
35J25
35J60

Keywords

Hardy potential
Ground state
Sign-changing solutions

Cited by (0)

Supported by NSFC (10871109).