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

TL;DR

This paper investigates positive solutions to a singularly perturbed nonlinear elliptic equation on Riemannian manifolds with boundary, revealing how the solutions relate to the manifold's topological features.

Contribution

It establishes a link between the number of solutions and the topological properties of the manifold, especially the Lusternik Schnirelmann category of the boundary.

Findings

Number of solutions depends on topological properties

Solutions are positive and satisfy Neumann boundary conditions

Topological invariants influence solution multiplicity

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 -{\epsilon}^2\Delta_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partial{\nu}=0 on \partial M 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.