Electron. J. Diff. Eqns., Vol. 2003(2003), No. 99, pp. 1-18.

Qualitative properties of solutions for quasi-linear elliptic equations

Zhenyi Zhao

Abstract:
For several classes of functions including the special case $f(u)=u^{p-1}-u^m$, $m greater than p-1 greater than 0$, 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 $u$ does not 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.

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Zhenyi Zhao
Department of Mathematical Sciences
Tsinghua University
Beijing, 100084, China
email: zhaozhenyi@tsinghua.org.cn

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