QUALITATIVE PROPERTIES OF SINGULAR SOLUTIONS OF CHOQUARD EQUATIONS

Our aim of this paper is to study qualitative properties of isolated singular solutions to Choquard equation -delta u + u = I-alpha[u(p)]u(q) + k delta(0 )in D'(R-N), (0.1) where p, q >= 1, N >= 2, alpha is an element of (0, N), k > 0, delta(0) is the Dirac mass concentrated at the origin and I-alpha[u(p)](x) = integral(RN)u(y)(p)/|x-y|(N-alpha) dy. Multiple proper-ties of very weak solutions of (0.1) are considered: (i) to obtain the existence of minimal solutions and extremal solutions for N = 2, which are derived in [8] when N >= 3; (ii) to analyze the stability of minimal solutions and the semi stability of extremal solutions; (iii) to derive a second solution by the Mountain Pass theorem when q = p - 1 and N = 2, 3; (iv) to obtain the radial symmetry of the positive singular solutions by the method of moving planes.