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 where 0 < γ 1 and d 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (γ) are radially symmetric about some point x0 d and derive the explicit form for u in the á 2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.