STABLE SOLUTIONS TO THE STATIC CHOQUARD EQUATION

This paper is concerned with the static Choquard equation -Delta u = (1/vertical bar x vertical bar(N-alpha) * vertical bar u vertical bar(p))vertical bar u vertical bar(p-2)u in R-N, where N, p > 2 and max {0, N - 4} < alpha < N. We prove that if u is an element of C-1(R-N) is a stable weak solution of the equation, then u equivalent to 0. This phenomenon is quite different from that of the local Lane-Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and alpha not equal 2.