Some Liouville Theorems for the p-Laplacian

TL;DR

This paper establishes several Liouville theorems for the p-Laplacian in Euclidean space, identifying conditions under which solutions must be trivial or constant, depending on growth, boundedness, and nonlinearity parameters.

Contribution

The paper provides new non-existence and constancy results for solutions of p-Laplacian equations with specific growth and boundedness conditions, extending classical Liouville theorems.

Findings

Non-existence of nontrivial solutions for certain q and γ ranges.

Solutions are constant when bounded below or in certain dimensional cases.

Solutions must be trivial when q > p-1 under specified conditions.

Abstract

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