A Liouville-type theorem in conformally invariant equations

TL;DR

This paper proves a Liouville-type theorem for conformally invariant equations in two dimensions, showing non-existence of solutions under certain conditions on the function K(x), extending previous results and considering higher order cases.

Contribution

It establishes a new non-existence result for solutions of a class of conformally invariant equations with polynomially constrained K(x), including higher order cases with additional assumptions.

Findings

No solutions for bounded u with finite energy when K(x) satisfies the cone condition.

Extends previous results to polynomial K(x) with x·∇K ≤ 0.

Addresses higher order equations with specific behavior of Δu at infinity.

Abstract

Given a smooth function $K(x)$ satisfying a polynomially cone condition and $x\cdot\nabla K\leq 0$, we prove that there is no solution $u\in C^\infty(\mathbb{R}^2)$ of the equation $$-\Delta u=K(x)e^{2u}\quad \mathrm{on}\;\mathbb{R}^2$$ with $u\leq C$ and $\int_{\mathbb{R}^2}|K(x)|e^{2u}d x<+\infty$. As a consequence, there is no such solution if $K(x)$ is a non-constant polynomial with $x\cdot\nabla K\leq 0$. The latter result already includes a result of Struwe(JEMS 2020) as a particular case. Higher order cases are set up with additional assumption on the behavior of $\Delta u$ near infinity.