Positive solutions of \(\Delta_ g u = u^ q + Su\) singular at submanifolds with boundary. (English) Zbl 0830.35035

Finn, David L.

Indiana Univ. Math. J. 43, No. 4, 1359-1397 (1994).

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

Reviewer: David L.Finn (Boston)

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