On the number of nodal solutions to a singularly perturbed Neumann problem

We show that for epsilon small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem epsilon(2) Delta u - u + f(u) = 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega where Omega is a bounded and smooth domain in R-2 and f grows superlinearly. ( A typical f(u) is f(u) = alpha(1)u(+)(p) - alpha(2)u(-)(q), alpha(1), alpha(2) > 0, p, q > 1.) More precisely, for any positive integer K, there exists epsilon(K) > 0 such that for 0 < epsilon < epsilon(K), the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K + 1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on partial derivative Omega. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.