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

We consider the problem -epsilon(2) Delta u + u = vertical bar u vertical bar(p-1) u in Omega, partial derivative u/partial derivative nu = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, 1 < p < +infinity if N = 2, 1 < p < (N + 2)/(N - 2) if N >= 3 and epsilon is a parameter. We show that if the mean curvature of partial derivative Omega is not constant then, for epsilon small enough, such a problem has always a nodal solution u(epsilon) with one positive peak xi(epsilon)(1) and one negative peak xi(epsilon)(2) on the boundary. Moreover, H(xi(epsilon)(1)) and H(xi(epsilon)(2)) converge to max(partial derivative Omega) H and min(partial derivative Omega) H, respectively, as epsilon goes to zero. Here, H denotes the mean curvature of partial derivative Omega. Moreover, if Omega is a ball and N >= 3, we prove that for epsilon small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily many negative peaks on the boundary.