Combinatorics

Physics

Hartree

Nonlinear system

Pure mathematics

Mathematics

Mathematical physics

Biology

Sobolev space

Harmonic

Mathematical analysis

Ecology

Type (biology)

Quantum mechanics

Harmonic function

Proceedings of the Royal Society of Edinburgh Section A Mathematics

In 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 &amp;lt; γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫 γ ) are radially symmetric about some point x 0 ∈ ℝ d and derive the explicit form for u in the Ḣ 2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &amp;lt; γ &amp;lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

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