Abstract
Let $\Omega$ be a bounded domain in $\mathbb{R}^2$. The study, begun in Keady[l981] and Keady and Kloeden[1984] of the boundary-value problem, for (\lambda/k, \psi) \[-\Delta \psi \in \lambda H (\psi - k) \hspace{.1in} in \hspace{.1in} \Omega \subset \mathbb{R}^2, \psi = 0 \hspace{.1in} on \hspace{.1in} \partial \Omega, \] is continued. Here $\Delta$ denotes the Laplacian, $H$ is the Heaviside step function and one of $\lambda$ or $k$ is a given positive constant. The solutions considered always have $\psi \gt 0$ in $\Omega$ and $\lambda/k \gt 0$, and have cores \[ A = {(x,y) \in \Omega | \psi(x,y) \gt k} \] In the special case $\Omega = B(O,R)$ , a disc, the explicit exact solutions are available. They satisfy \[ (\ast) \hspace{2in} (\psi_{m} - k)/k \rightarrow 0 \hspace{.1in} as \hspace{.1in} area(A) \rightarrow 0 , \] where $\psi_m$ is the maximum of $\psi$ over $\Omega$. Here (\ast) will be established for other domains. An adaptation of the maximum principles of Gidas, Ni and Nirenberg [1979] is an important step in establishing the above result.
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