Positive solutions of $\Delta u+u^p = 0$ whose singular set is a manifold with boundary

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The aim of this paper is to prove the existence of weak solutions to the equation $\Delta u+u^p = 0$ , with $n \geq 4$ , which are positive in a domain $\Omega \subset \mathbb{R}^n$ and which are singular along a k-dimensional submanifold with smooth boundary. Here, the exponent p is required to lie in the interval $[\frac{n-k}{n-2-k},\frac{n-k+2}{n-2-k})$ , where $1 \leq k < n-2$ is the dimension of the singular set. In the particular case where $p = \frac{n+2}{n-2}$ and $\Omega = \mathbb{R}^n$ , solutions correspond to solutions of the singular Yamabe problem.

Received: 7 October 2001 / Accepted: 7 March 2002 / Published online: 6 August 2002