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A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. (English) Zbl 0869.35040
Authors’ summary: The aim of this paper is to prove the existence of weak solutions to the equation $$\Delta u+u^p=0$$ which are positive in a domain $$\Omega\subset\mathbb{R}^N$$, vanish at the boundary, and have prescribed isolated singularities. The exponent $$p$$ is required to lie in the interval $$(N/(N-2),(N+2)/(N-2))$$. We also prove the existence of solutions to the equation $$\Delta u+ u^p=0$$ which are positive in a domain $$\Omega\subset\mathbb{R}^n$$ and which are singular along arbitrary smooth $$k$$-dimensional submanifolds in the interior of these domains, provided $$p$$ lies in the interval $$((n-k)/(n-k-2),(n-k+2)/(n-k-2))$$. A particular case is when $$p=(n+2)/(n-2)$$, in which case solutions correspond to solutions of the singular Yamabe problem. The method used here is a mixture of different ingredients used by both authors in their separate constructions of solutions to the singular Yamabe problem, along with a new set of scaling techniques.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35C20 Asymptotic expansions of solutions to PDEs
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