Electronic Journal of Differential Equations (Sep 2003)
Qualitative properties of solutions for quasi-linear elliptic equations
Abstract
For several classes of functions including the special case $f(u)=u^{p-1}-u^m$, $m>p-1>0$, we obtain Liouville type, boundedness and symmetry results for solutions of the non-linear $p$-Laplacian problem $-Delta_p u=f(u)$ defined on the whole space $mathbb{R}^n$. Suppose $u in C^2(mathbb{R}^n)$ is a solution. We have that either (1) if $u$ doesn't change sign, then $u$ is a constant (hence, $uequiv 1$ or $uequiv 0$ or $uequiv-1$); or (2) if $u$ changes sign, then $uin L^{infty}(mathbb{R}^n)$, moreover $|u|0$ on $mathbb{R}^n$ and the level set $u^{-1}(0)$ lies on one side of a hyperplane and touches that hyperplane, i.e., there exists $ u in S^{n-1}$ and $x_{0}in u^{-1}(0)$ such that $ u cdot (x-x_0)geq 0$ for all $xin u^{-1}(0)$, then $u$ depends on one variable only (in the direction of $ u$).
