On the existence of some new positive interior spike solutions to a semilinear Neumann problem

In this paper we are concerned with the following Neumann problem {epsilon(2)Delta u - u + f(u) = 0, u > 0 in Omega, partial derivative u/partial derivative nu = 0 on partial derivative Omega, where epsilon is a small positive parameter, f is a superlinear and subcritical nonlinearity, Omega is a smooth and bounded domain in R(N). Solutions with multiple boundary peaks have been established for this problem. It has also been proved that for any integer k there exists all interior k-peak solution which concentrates. as a epsilon -> 0(+). at k sphere packing points in Omega. In this paper we prove the existence of a second interior k-peak solution provided that k is large enough, and we conjecture that its peaks are located along a straight line. Moreover, when Omega is a two-dimensional strictly convex domain, we also construct a third interior k-peak solution provided that k is large enough, whose peaks are aligned on a closed curve near partial derivative Omega. (C) 2009 Elsevier Inc. All rights reserved.