On Yamabe-type problems on Riemannian manifolds with boundary

Marco Ghimenti, Anna Maria Micheletti and Angela Pistoia

Vol. 284 (2016), No. 1, 79–102

DOI: 10.2140/pjm.2016.284.79

Abstract

Let $\left(M,g\right)$ be an $n$-dimensional compact Riemannian manifold with boundary. We consider the Yamabe-type problem

$$\left\{\begin{array}{cc}-{\Delta }_{g}u+au=0\hfill & \text{ on }M,\hfill \\ {\partial }_{\nu }u+\frac{n-2}{2}bu=\left(n-2\right)u^{n∕\left(n-2\right)\pm \epsilon }\hfill & \text{ on }\partial M,\hfill \end{array}\right.$$

where $a\in C^1\left(M\right)$, $b\in C^1\left(\partial M\right)$, $\nu$ is the outward pointing unit normal to $\partial M$, ${\Delta }_{g}u:={\text{ div}}_g{\nabla }_{g}u$, and $\epsilon$ is a small positive parameter. We build solutions which blow up at a point of the boundary as $\epsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g$, where $H_g$ is the boundary mean curvature.

Keywords

Yamabe problem, blowing-up solutions, compactness

Mathematical Subject Classification 2010

Primary: 35J20, 58J05

Milestones

Received: 29 May 2015

Revised: 17 January 2016

Accepted: 31 March 2016

Published: 10 July 2016