zbMATH — the first resource for mathematics

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 \(n - 2 - 2/(q - 1)\).

35J60 Nonlinear elliptic equations
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI