Singularly perturbed nonlinear Neumann problems with a general nonlinearity

Let Ω be a bounded domain in Rn, n⩾3, with the boundary ∂Ω∈C3. We consider the following singularly perturbed nonlinear elliptic problem on Ωε2Δu−u+f(u)=0,u>0on Ω,∂u∂ν=0on ∂Ω, where ν is an exterior normal to ∂Ω and a nonlinearity f of subcritical growth. Under rather strong conditions on f, it has been known that for small ε>0, there exists a solution uε of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on ∂Ω as ε→0. In this paper, we obtain the same result under some conditions on f (Berestycki-Lions conditions), which we believe to be almost optimal.