Multiple boundary peak solutions for some singularly perturbed Neumann problems

We consider the problem {epsilon(2)Delta u - u + f(u) = 0 in Omega, u > 0 in Omega, partial derivative u/partial derivative nu = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, epsilon > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as epsilon approaches zero, at a critical point of the mean curvature function H(P), P is an element of partial derivative Omega. It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P). In this paper, we prove that for any fixed positive integer K there exist boundary K-peak solutions at a local minimum point of H(P). This implies that for any smooth and bounded domain there always exist boundary K-peak solutions.