Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight

Let p > 0 and (-Delta)(s) is the fractional Laplacian with 0 < s < 1. The purpose of this paper is to establish a classification result for positive stable solutions to a fractional singular elliptic equation with weight (-Delta)(s)u = -h(x)u(-p) in R-N. Here N > 2s and h is a nonnegative, continuous function satisfying h(x) >= C vertical bar x vertical bar(a), a >= 0, when vertical bar x vertical bar large. We prove the nonexistence of positive stable solutions of this equation under the condition N < 2s + 2(a +2s)/p + 1 (p + root p(2) + p) or equivalently p > p(c (N, s, a),) where p(c) (N, s, a) = {(N-2s)(2)-2(N+a) (a+2s)+2 root(a + 2s)(3)(2N-2s+a)/(N-2s) (10s+4a-N) if N < 10s + 4a +infinity if N >= 10s + 4a