Multiple interior peak solutions for some singularly perturbed Neumann problems

We consider the problem [GRAPHICS] 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 that concentrate, 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 proved that this equation has single interior spike solutions at a local maximum point of the distance function d(P, partial derivative Omega), P is an element of Omega. In this paper, we prove the existence of interior K- peak (K greater than or equal to 2) solutions at the local maximum points of the following function phi(P-1, P-2, ..., P-K) = min(i,k,l=1, ..., K; k not equal l) (d(P-i, partial derivative Omega), 1/2\P-k - P-l\). We first use the Liapunov-Schmidt reduction method to reduce the problem to a finite dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function phi(P-1, ..., P-K) appears naturally in the asymptotic expansion of the energy functional. (C) 1999 Academic Press.