Differential and Integral Equations

Coincidence sets in semilinear elliptic problems of logistic type

Shingo Takeuchi

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Abstract

This paper concerns the formation of a coincidence set for the positive solution of an equation of the type: $-{\varepsilon} \Delta u=u|a(x)-u|^{\theta} {\operatorname{sgn}} (a(x)-u)$, where ${\varepsilon}$ is a positive parameter, $0 < \theta < 1$ and $a(x)$ is a positive continuous function. Suppose that $\Delta a=0$ in an open subset ${\Omega}_0 \subset {\Omega}$. The positive solution converges to $a(x)$ uniformly on any compact subset of $\Omega$ as ${\varepsilon} \to 0$. It is proved that when ${\varepsilon}$ is sufficiently small, the solution coincides with $a(x)$ somewhere in ${\Omega}_0$ and the coincidence set converges to ${\Omega}_0$ in the Hausdorff distance with the order of $\sqrt{{\varepsilon}}$ as ${\varepsilon} \to 0$. The proof relies on the comparison principle with suitable local comparison functions.

Article information

Source
Differential Integral Equations Volume 20, Number 9 (2007), 1075-1080.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039312

Mathematical Reviews number (MathSciNet)
MR2349381

Zentralblatt MATH identifier
1212.35141

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B25: Singular perturbations 35J25: Boundary value problems for second-order elliptic equations

Citation

Takeuchi, Shingo. Coincidence sets in semilinear elliptic problems of logistic type. Differential Integral Equations 20 (2007), no. 9, 1075--1080. https://projecteuclid.org/euclid.die/1356039312.


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