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.
Categories: math.AP