Topological Methods in Nonlinear Analysis

Positive solutions of singularly perturbed nonlinear elliptic problem on Riemannian manifolds with boundary

Marco Ghimenti and Anna M. Micheletti

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Abstract

Let $(M,g)$ be a smooth connected compact Riemannian manifold of finite dimension $n\geq 2$\ with a smooth boundary $\partial M$. We consider the problem $$ \begin{cases} -\varepsilon ^{2}\Delta _{g}u+u=|u|^{p-2}u,\quad u> 0 &\text{ on }M,\\ \displaystyle \frac{\partial u}{\partial \nu }=0 & \text{on }\partial M, \end{cases} $$ where $\nu $ is an exterior normal to $\partial M$.

The number of solutions of this problem depends on the topological properties of the manifold. In particular we consider the Lusternik Schnirelmann category of the boundary.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 35, Number 2 (2010), 319-337.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461251010

Mathematical Reviews number (MathSciNet)
MR2676820

Zentralblatt MATH identifier
1204.58017

Citation

Ghimenti, Marco; Micheletti, Anna M. Positive solutions of singularly perturbed nonlinear elliptic problem on Riemannian manifolds with boundary. Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 319--337. https://projecteuclid.org/euclid.tmna/1461251010


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