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 ε2Δu−u+f(u)=0, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where ε is a small positive parameter, f is a superlinear and subcritical nonlinearity, Ω is a smooth and bounded domain in RN. Solutions with multiple boundary peaks have been established for this problem. It has also been proved that for any integer k there exists an interior k-peak solution which concentrates, as ε→0+, at k sphere packing points in Ω. 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 Ω 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 ∂Ω.