A Liouville-type theorem in conformally invariant equations

Given a smooth function $K(x)$ satisfying a polynomially cone condition and $x\cdot \mathrm{\nabla }K\le 0$, we prove that there is no solution $u\in C^{\mathrm{\infty }}({\mathbb{R}}^2)$ of the equation $$-\mathrm{\Delta }u=K(x)e^{2u}\mathrm{o}\mathrm{n}{\mathbb{R}}^2$$ with $u\le C$ and ${\int }_{{\mathbb{R}}^2}\mathrm{\mid }K(x)\mathrm{\mid }{e}^{2u}dx<+\mathrm{\infty }$. As a consequence, there is no such solution if $K(x)$ is a non-constant polynomial with $x\cdot \mathrm{\nabla }K\le 0$. 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 $\mathrm{\Delta }u$ near infinity.