Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 99, pp. 1-18.
Title: Qualitative properties of solutions for quasi-linear elliptic equations
Author: Zhenyi Zhao (Tsinghua Univ., Beijing, China)
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, $u\equiv 1$ or $u\equiv 0$ or $u\equiv-1$); or
(2) if $u$ changes sign, then $u\in L^{\infty}(\mathbb{R}^n)$,
moreover $|u|<1$ on $\mathbb{R}^n$; or
(3) if $|Du|>0$ 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 $\nu \in S^{n-1}$ and $x_{0}\in u^{-1}(0)$ such that
$\nu \cdot (x-x_0)\geq 0$ for all $x\in u^{-1}(0)$, then $u$ depends
on one variable only (in the direction of $\nu$).
Submitted May 29, 2003. Published September 25, 2003.
Math Subject Classifications: 35J15, 35J25, 35J60.
Key Words: Quasi-linear elliptic equations; comparison Principle;
boundary blow-up solutions; moving plane method; sliding method;
symmetry of solution.