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

Abstract

In this paper we are concerned with the following Neumann problem { ε 2 Δ u − 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 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 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.