Some nonexistence results for positive solutions of elliptic equations in unbounded domains

Abstract

We prove some Liouville type theorems for positive solutions of semilinear elliptic equations in the whole space ${\mathbb{R}}^N$, $N\ge 3$, and in the half space ${\mathbb{R}}_{+}^N$ with different boundary conditions, using the technique based on the Kelvin transform and the Alexandrov-Serrin method of moving hyperplanes. In particular we get new nonexistence results for elliptic problems in half spaces satisfying mixed (Dirichlet-Neumann) boundary conditions.