A strong comparison principle for positive solutions of degenerate elliptic equations

Abstract

A strong comparison principle (SCP, for brevity) is obtained for nonnegative weak solutions $u\in W_0^{1,p}(\Omega)$ of the following class of quasilinear elliptic boundary value problems, \begin{equation} -\mbox{div }( {\bf a}(x,\nabla u) ) - b(x,u) = f(x) \;\hbox{ in } \Omega ; \quad u = 0 \;\hbox{ on } \partial\Omega . \tag*{(P)} \end{equation} Here, $p\in (1,\infty)$ is a given number, $\Omega$ is a bounded domain in $\mathbb R^N$ with a connected $C^2$-boundary, ${\bf a}(x,\nabla u)$ and $b(x,u)$ are slightly more general than the functions $a_0(x) |\nabla u|^{p-2}\nabla u$ and $b_0(x) |u|^{p-2} u$, respectively, with $a_0\geq $const$ > 0$ and $b_0\geq 0$ in $L^\infty (\Omega)$, and $0\leq f\in L^\infty (\Omega)$. Validity of the SCP is investigated also in the case when $b_0\leq 0$ depending upon whether $p\leq 2$ or $p>2$. The methods of proofs are new.