Differential and Integral Equations

A strong comparison principle for positive solutions of degenerate elliptic equations

Mabel Cuesta and Peter Takáč

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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.

Article information

Source
Differential Integral Equations Volume 13, Number 4-6 (2000), 721-746.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356061247

Mathematical Reviews number (MathSciNet)
MR1750048

Zentralblatt MATH identifier
0973.35077

Subjects
Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.
Secondary: 34B15: Nonlinear boundary value problems 34C11: Growth, boundedness 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 35J70: Degenerate elliptic equations

Citation

Cuesta, Mabel; Takáč, Peter. A strong comparison principle for positive solutions of degenerate elliptic equations. Differential Integral Equations 13 (2000), no. 4-6, 721--746. https://projecteuclid.org/euclid.die/1356061247.


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