Some Liouville Theorems for the p-Laplacian

Abstract

We present several Liouville type results for the $p$-Laplacian in ${\mathbb{R}}^N$. Suppose that $h$ is a nonnegative regular function such that $$h(x)=a|x{|}^{\gamma }\mathrm{for}|x|\mathrm{large},a>0\mathrm{and}\gamma >-p.$$ We obtain the following non -existence result: 1) Suppose that $N>p>1$, and $u\in W_{loc}^{1,p}({\mathbb{R}}^N)\cap \mathcal{C}({\mathbb{R}}^N)$ is a nonnegative weak solution of $ - {\rm div} (|\nabla u|^{p-2 }\nabla u) \geq h(x) u^q \;\;\mbox{in }\; \R^N $ . Suppose that $p-1<q\le \frac{(N+\gamma )(p-1)}{N-p}$ then $u\equiv 0$. 2) Let $N\le p$. If $u\in W_{loc}^{1,p}({\mathbb{R}}^N)\cap \mathcal{C}({\mathbb{R}}^N)$ is a weak solution bounded below of $-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)\ge 0$ in ${\mathbb{R}}^N$ then $u$ is constant. 3) Let $N>p$ if $u$ is bounded from below and $-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)=0$ in ${\mathbb{R}}^N$ then $u$ is constant. 4)If $-{\Delta }_{p}u+h(x)u^q\le 0,$. If $q>p-1$, then $u\equiv 0$.

19 pages

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