Liouville type theorems for poly-harmonic Navier problems

Abstract

In this paper we consider the following semi-linear poly-harmonic equation with Navier boundary conditions on the half space $R^n_+$: \begin{equation} \left\{\begin{array}{l} (-\triangle)^{\frac{\alpha}{2}} u=u^p, \:\:\: \:\:\:\:\:\ \:\: \ \:\:\:\: \mbox{in}\, R^n_+,\\ u=-\triangle u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0, \mbox{on} \partial R^n_+, \end{array} \right. \label{phe1} \end{equation} where $\alpha$ is any even number between $0$ and $n$, and $p>1$. &nbsp First we prove that (1) is equivalent to the following integral equation \begin{equation} u(x)=\int_{R^n_+}G(x,y,\alpha) u^p(y)dy,\,\,\,\,\, x\in\,R^n_+, \label{ie0} \end{equation} under some very mild growth condition, where $G(x, y,\alpha)$ is the Green's function associated with the same Navier boundary conditions on the half-space . &nbsp Then by combining the method of moving planes in integral forms with a certain type of Kelvin transform, we obtain the non-existence of positive solutions for integral equation (2) in both subcritical and critical cases under only local integrability conditions. This remarkably weaken the global integrability assumptions on solutions in paper [3]. Our results on integral equation (2) are valid for all real values $\alpha$ between $0$ and $n$. &nbsp Finally, we establish a Liouville type theorem for PDE (1), and this generalizes Guo and Liu's result [21] by significantly weaken the growth conditions on the solutions.