Liouville results for $m$-Laplace equations of Lane–Emden–Fowler type

Abstract

We consider sign changing solutions of the equation $-{\mathrm{\Delta }}_m(u)=|u{|}^{p-1}u$ in possibly unbounded domains or in ${\mathbb{R}}^N$. We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The results hold true for $m>2$ and $m-1<p<p_c(N,m)$. Here $p_c(N,m)$ is a new critical exponent, which is infinity in low dimension and is always larger than the classical critical one.