Singularly perturbed nonlinear Neumann problems under the conditions of Berestycki and Lions

Abstract

Let Ω be a bounded domain in R N with the boundary ∂ Ω ∈ C 3 . We consider the following singularly perturbed nonlinear elliptic problem on Ω, ε 2 Δ v − v + f ( v ) = 0 , v > 0 on Ω , ∂ v ∂ ν = 0 on ∂ Ω , where ν is the exterior normal to ∂ Ω and the nonlinearity f is of subcritical growth. It has been known that under Berestycki and Lions conditions for f ∈ C 1 ( R ) and N ⩾ 3 , there exists a solution v ε of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂ Ω for small ε > 0 . In this paper, we extend the previous result for f ∈ C 0 ( R ) and N ⩾ 2 .