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

Let Ω   be a bounded domain in RNRN with the boundary ∂Ω∈C3∂Ω∈C3. We consider the following singularly perturbed nonlinear elliptic problem on Ω,ε2Δv−v+f(v)=0,v>0on Ω,∂v∂ν=0on ∂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∈C1(R)f∈C1(R) and N⩾3N⩾3, there exists a solution vεvε of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂Ω   for small ε>0ε>0. In this paper, we extend the previous result for f∈C0(R)f∈C0(R) and N⩾2N⩾2.