Nodal Clustered Solutions for Some Singularly Perturbed Neumann Problems

In this paper we are concerned with the following Neumann problem [image omitted] where epsilon is a small positive parameter, f is an odd superlinear and subcritical nonlinearity, is a bounded C4 domain in N without any symmetry assumption. Denoting by H(P), P , the mean curvature of the boundary, it is known that this problem has positive multiple boundary peak solutions with each peak concentrating at a different critical point of H or with all the peaks approaching a local minimum point of H. In this paper we assume that H has a nondegenerate maximum point P0 and we show that there exists a -peak solution with mixed positive and negative peaks concentrating at P0.