Singularly perturbed nonlinear Neumann problems with a general nonlinearity

Abstract

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 Ωε2 Δ u - u + f (u) = 0, u > 0 on Ω, frac(∂ u, ∂ ν) = 0 on ∂ 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. © 2008 Elsevier Inc. All rights reserved.