ON THE EFFECT OF DOMAIN GEOMETRY ON THE EXISTENCE OF NODAL SOLUTIONS IN SINGULAR PERTURBATIONS PROBLEMS

The existence of a nodal solution with least energy is established for the problem -epsilon(2)Delta u + u = \u\(p-1) in Omega u is an elemen t of H-0(1)(Omega) where Omega is a bounded domain, 2 < p < (N + 2)/(N - 2) for N greater than or equal to 3, 2 < p < infinity for N = 2. It is shown that such a solution has, for small epsilon, exactly one pos itive and one negative peaks, and that the peak points converge, as ep silon --> 0, to two distinct points P-1, P-2 of Omega, whose locations depend on the geometry of Omega.