Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity (P-gamma) Delta(2)u = (1/vertical bar x vertical bar(8) * vertical bar u vertical bar(2)) u(gamma), x is an element of R-d , where 0 < gamma and d >= 9. By the By applying the method of moving planes, we prove that nonnegative classical solutions u to (P.) are radially symmetric about some point x0. Rd and derive the explicit form for u in the. H 2 critical case. = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 <. < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.