Existence and nonexistence of solutions to elliptic equations involving the Hardy potential

The purpose of this paper is to study the nonexistence of nonnegative super solutions to the problem(0.1)(−Δ)αu+μ|x|2αu≥QupinRN∖K, where α∈(0,1], μ∈R, p>0, K is a compact set in RN with N≥1 and Q is a potential in RN∖K satisfying that liminf|x|→+∞Q(x)|x|γ>0 for some γ<2α. When α=1, (−Δ)α is the Laplacian operator, and when α∈(0,1), it is the fractional Laplacian which is a typical nonlocal operator. In this paper, we find the critical exponent p⁎>1 depending on α,μ and γ such that problem (0.1) has no nontrivial nonnegative super solutions for 00, p>0 and Q(x)=(1+|x|)−γ with γ∈(0,2α).