Nodal clustered solutions for some singularly perturbed Neumann problems

In this paper we are concerned with the following Neumann problem where ϵ is a small positive parameter, f is an odd superlinear and subcritical nonlinearity, Ω is a bounded 4 domain in N without any symmetry assumption. Denoting by (P), P ∂Ω, 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 or with all the peaks approaching a local minimum point of . In this paper we assume that has a nondegenerate maximum point P 0 ∂Ω and we show that there exists a ℓ-peak solution with mixed positive and negative peaks concentrating at P 0.