Nodal Solutions for a Singularly Perturbed Nonlinear Elliptic Problem on Riemannian Manifolds

Let (M, g) be a smooth compact, Riemannian N-manifold, N >= 2. We show that if the scalar curvature of g is not constant, the problem -epsilon(2)Delta(g)u + u = u(p-1) in M has a positive solution with two positive peaks xi(epsilon)(1) and xi(epsilon)(2) and a sign changing solution with one positive peak eta(epsilon)(1) and one negative peak eta(epsilon)(2), such that as epsilon goes to zero Scal(g)(xi(epsilon)(1)), Scal(g)(eta(epsilon)(1)) -> min(epsilon is an element of M) Scal(g)(xi) and Scal(g)(xi(epsilon)(2)), Scal(g)(eta(epsilon)(2)) -> max(epsilon is an element of M) Scal(g)(xi). Here p > 2 if N = 2 and 2 < p < 2* = 2N/N-2 if N >= 3.