Exact multiplicity results for a singularly perturbed Neumann problem

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In this paper we study the number of the boundary single peak solutions of the problem align* cases -ε² Δu + u = uᵖ, &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.