On stable solutions of a weighted elliptic equation involving the fractional Laplacian

In this paper, we study the following fractional Choquard equation with weight (-Delta)(s)u = (1/vertical bar x vertical bar(N-alpha) * h(x)vertical bar u vertical bar(p)) h(x)vertical bar u vertical bar(p-2)u in R-N, where 0 < s < 1, N > 2s, p > 2, alpha > 0 and h is a positive weight function satisfying h(x) >= C vertical bar x vertical bar(a) at infinity, for some a >= 0. We establish, in this paper, a Liouville type theorem saying that if max (N - 4s - 2a, 0) < alpha < N, then the above equation has no nontrivial stable solution. Our result, in particular, extends the result in [Le, Phuong. Bull. Aust. Math. Soc. 102 (2020), no. 3, 471-478.] from the Laplace operator to the fractional Laplacian.