Classifications of nonnegative solutions to some elliptic problems. (Q1854090)

The authors study the problem \(\Delta u = 0\) (or \(\Delta u = u^p\)) with \(u\geq 0\) in \(H:=\mathbb R^{n-1}\times \mathbb R_+\), \(n\geq 2\), \({\partial u}/{\partial x_n}= u^q\) on \(\partial H\) and show that \(u=ax_n +b\) with \(a=b^q>0\) (or \(u\equiv 0\)) under the assumption \(q>1\) (or \(p,q>1\)). The authors use the moving plane method and the Strong Maximum Principle to prove the result first in a supercritical case \(q>n/(n-2)\) and \(n\geq 3\). Consequently in a subritical case or \(q>1\) and \(n=2\), they consider the original problem in a high dimensional space setting to obtain a supercritical problem again.