Classification of solutions to an $n^{\text{th}}$ order conformally invariant elliptic equation on ${\mathbb{R}}^n$ with nonlocal nonlinearity

Abstract

This paper concerns a conformally invariant elliptic problem on ${\mathbb{R}}^n$ driven by $(-\Delta {)}^{n/2}$ that has a nonlocal exponential nonlinearity of Choquard type. The problem under consideration is a nonlocal generalization of the constant $Q$-curvature problem on ${\mathbb{R}}^n$. We classify the asymptotic behavior at infinity of all solutions that satisfy a suitable integrability assumption. Under a growth restriction at infinity and a lower bound on the energy we provide an explicit classification for solutions. The classification is heuristically consistent with the classification of the corresponding local problem.