A Liouville type theorem for poly-harmonic Dirichlet problems in a half space (Q409628)

The authors consider the following Dirichlet problem for poly-harmonic operators on a half space \({\mathbb R}^n_{+}\): NEWLINE\[NEWLINE \left\{ \begin{aligned} &(-\Delta)^m u = u^p \qquad \qquad \qquad \qquad \qquad \qquad \text{in} \, {\mathbb R}^n_{+},\\ &u = \frac{\partial u}{\partial x_n} = \frac{\partial^2 u}{\partial x_n^2} = \cdots = \frac{\partial^{m-1} u}{\partial x_n^m-1} = 0 \quad \,\, \text{on}\, {\partial {\mathbb R}^n_{+}}, \end{aligned}\right.\tag{1} NEWLINE\]NEWLINE where \(m\) is any positive integer, \(2m < n\), and \(1 < p \leq \frac{n+2m}{n-2m}\).NEWLINENEWLINEThis problem has been considered in [\textit{W. Reichelt} and \textit{T. Weth}, J. Differ. Equations 248, No. 7, 1866--1878 (2010; Zbl 1185.35066)]. They proved that there are no bounded classical solutions. In this paper, the authors removes their boundedness assumptions on \(u\) and all its derivatives, and under very mild growth conditions, the authors show that problem (1) is equivalent to the integral equation NEWLINE\[NEWLINE u(x) = \int_{{\mathbb R}^n_+} G(x, y) u^p\, dy,\tag{2} NEWLINE\]NEWLINE where \(G(x, y)\) is the Green's function on the half space. Using this equivalence and showing that there is no positive solution for integral equation (2), the authors prove that there is no positive solution for the equation (1) in both subcritical and critical cases.