LIOUVILLE THEOREMS AND CLASSIFICATION RESULTS FOR A NONLOCAL SCHRODINGER EQUATION

In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation -Delta u = pu(p-1)(vertical bar x vertical bar(2-n)*u(p)), u > 0 in R-n, where n >= 3 and p >= 1. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when 1 <= p < n+2/n-2 by means of the method of moving planes to the following system { -Delta u = root pu(p-1)v, u > 0 in R-n, -Delta v = root pu(p), v > 0 in R-n. When p = n+2/n-2, all the positive solutions can be classified as u(x) = c(t/t(2) + vertical bar x - x*vertical bar(2))(n-2/2) with the help of an integral system involving the Newton potential, where c, t are positive constants, and x* is an element of R-n. In addition, we also give other equivalent conditions to classify those positive solutions. When p > n+2/n-2, by the shooting method and the Pohozaev identity, we find radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate 2/p-1. Finally, we point out that the equation has positive stable solutions if and only if p >= 1 + 4/n-4-2 root n-1.