## Differential and Integral Equations

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

### Coincidence sets in semilinear elliptic problems of logistic type

#### 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