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

Abstract

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity$$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 < γ ⩽ 1 andd⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutionsuto (𝒫γ) are radially symmetric about some pointx0∈ ℝdand derive the explicit form foruin theḢ2critical 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.