Existence of solutions for a fractional elliptic problem with critical Sobolev-Hardy nonlinearities in R^N

Abstract

In this article, we study the fractional elliptic equation with critical Sobolev-Hardy nonlinearity (-∆)α u + α(x)u = |u|2*s - 2u/|x|s + k(x)|u|q-2u, u ∈ Hα (ℝN), where 2 < q < 2*, 0 < α < 1, N > 4α, 0 < s < 2α, 2*s = 2(N - s)/(N - 2α) is the critical Sobolev-Hardy exponent, 2* = 2N/(N - 2α) is the critical Sobolev exponent, α(x), k(x) ∈ C(ℝN). Through a compactness analysis of the functional associated, we obtain the existence of positive solutions under certain assumptions on α(x), k(x).