Exact multiplicity results for a singularly perturbed Neumann problem

2012/11/02 by Massimo Grossi, Grossi, Massimo, Sérgio Neves +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1211.0569

arxiv created 2012/11/02 · arxiv updated 2012/11/06

Abstract

In this paper we study the number of the boundary single peak solutions of the problem align* cases -ε² Δu + u = u^p, &inΩu > 0, &inΩ(∂ u)/(∂ ν) = 0,& on∂ Ωcases align* for ε 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.

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