A note on boundary blow-up problem of $\Delta u=u^p$
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Assume that $\Omega$ is a bounded domain in $\Real^n$ with $n\ge 2$. We study positive solutions to the problem, $\Delta u=u^p$ in $\Omega$, $u(x)\to\infty$ as $x\to\partial\Omega$, where $p>1$. Such solutions are called boundary blow-up solutions of $\Delta u=u^p$. We show that a boundary blow-up solution exists in any bounded domain if $1<p<\frac{n}{n-2}$. In particular, when $n=2$, there exists a boundary blow-up solution to $\Delta u=u^p$ for all $p\in(1,\infty)$. We also prove the uniqueness under the additional assumption that the domain satisfies the condition $\partial\Omega=\partial\overline{\Omega}$.
Abstract
Assume that $\Omega$ is a bounded domain in $\Real^n$ with $n\ge 2$. We study positive solutions to the problem, $\Delta u=u^p$ in $\Omega$, $u(x)\to\infty$ as $x\to\partial\Omega$, where $p>1$. Such solutions are called boundary blow-up solutions of $\Delta u=u^p$. We show that a boundary blow-up solution exists in any bounded domain if $1<p<\frac{n}{n-2}$. In particular, when $n=2$, there exists a boundary blow-up solution to $\Delta u=u^p$ for all $p\in(1,\infty)$. We also prove the uniqueness under the additional assumption that the domain satisfies the condition $\partial\Omega=\partial\overline{\Omega}$.
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