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On Yamabe type problems on Riemannian manifolds with boundary

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Abstract

Let $$(M,g)$$ be a $$n-$$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem $$\left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right.$$ 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.

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Journal
30 June 2015
1506.09105