On Yamabe type problems on Riemannian manifolds with boundary

TL;DR

This paper studies a boundary Yamabe problem on compact Riemannian manifolds, constructing solutions that blow up at boundary points as a small parameter approaches zero, influenced by boundary mean curvature.

Contribution

It introduces a method to construct boundary blow-up solutions for a Yamabe type problem involving a small parameter, highlighting the role of boundary mean curvature.

Findings

Solutions blow up at boundary points as epsilon approaches zero.

Blow-up behavior is governed by the function b - H_g.

The approach links geometric boundary data to solution behavior.

Abstract

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $\nu$ is the outward pointing unit normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$ is the boundary mean curvature.