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Positive solutions of $$\Delta_ g u = u^ q + Su$$ singular at submanifolds with boundary. (English) Zbl 0830.35035
We examine the existence of positive solutions to the nonlinear elliptic equation $$\Delta_gu = u^q + Su$$ on a compact Riemannian manifold $$(M,g)$$ with a prescribed singular set $$\Gamma$$. Further, we study the possible asymptotic behavior of singular solutions to this equation. The main results of this paper concern the case when the singular set $$\Gamma$$ is a smooth submanifold with boundary. In this case, we are able to show the existence of positive singular solutions to $$\Delta_gu = u^q + Su$$ with complex asymptotic behavior when the dimension of $$\Gamma$$ is greater than $$n - 2 - 2/(q - 1)$$.

##### MSC:
 35J60 Nonlinear elliptic equations 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
solutions with prescribed singular set; Yamabe problem
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