关键词: higher order equations, exterior domains, Liouville theorem, existence, supersolutions

Abstract: In this paper, we study the weighted higher order semilinear equation in an exterior domain $$\begin{equation*} (-\Delta)^{m} u=|x|^{\alpha}g(u) \quad \quad \text{in} \ \mathbb{R}^{N}\setminus B_{R_{0}}, \end{equation*}$$ where $N\geq1$, $m\geq2$ are integers, $\alpha>-2m$, $g$ is a continuous and nondecreasing function in $\left[ 0,+\infty\right) $ and positive in $\left( 0,+\infty\right) $, $ B_{R_{0}}$ is the ball of the radius $R_{0}$ centered at the origin. We prove that a positive supersolution of the problem which verifies $ (-\Delta )^{i}u > 0 $ in $\ \mathbb{R}^{N}\setminus B_{R_{0}}$ $(i=0,\cdots, m-1)$ exists if and only if $N>2m$ and $$\begin{equation*} \int_{0}^{\delta}\frac{g(t)}{t^{\frac{2(N-m)+\alpha}{N-2m}}}{\rm d}t<\infty, \end{equation*}$$ for some $\delta>0$. We further obtain some existence and nonexistence results for the positive solution to the Dirichlet problem when $g(u)=u^p$ with $p>1 $, by using the Pohozaev identity and an embedding lemma of radial Sobolev spaces.

Key words: higher order equations, exterior domains, Liouville theorem, existence, supersolutions