Exact multiplicity results for a singularly perturbed Neumann problem

TL;DR

This paper establishes precise counts of boundary single peak solutions for a singularly perturbed Neumann problem, considering small perturbations and boundary curvature conditions, including degenerate critical points.

Contribution

It provides exact multiplicity results for solutions of a singularly perturbed Neumann problem under boundary curvature assumptions, allowing for degenerate critical points.

Findings

Exact number of boundary solutions determined

Results valid for small perturbation parameter 

Includes cases with degenerate critical points

Abstract

In this paper we study the number of the boundary single peak solutions of the problem {align*} {cases} -\varepsilon^2 \Delta u + u = u^p, &\text{in}\Omega u > 0, &\text{in}\Omega \frac{\partial u}{\partial \nu} = 0,& \text{on}\partial \Omega {cases} {align*} for $\varepsilon$ small and $p$ subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.