On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem

We consider the problem where Ω is a bounded smooth domain in , 1  <  p< + ∞ if N = 2, if N ≥ 3 and ε is a parameter. We show that if the mean curvature of ∂Ω is not constant then, for ε small enough, such a problem has always a nodal solution u _ε with one positive peak and one negative peak on the boundary. Moreover, and converge to and , respectively, as ε goes to zero. Here, H denotes the mean curvature of ∂Ω. Moreover, if Ω is a ball and , we prove that for ε small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily many negative peaks on the boundary. The authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.