Differential and Integral Equations

Coincidence sets in semilinear elliptic problems of logistic type

Shingo Takeuchi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation