Locating the Peaks of Least-energy Solutions to a Quasilinear Elliptic Neumann Problem

Abstract. In this paper we study the shape of least-energy solutions to the quasilinear problem ε∆mu−u + f (u) = 0 with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that as ε → 0, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂Ω at the rate of o(ε) and this point on the boundary approaches to a point where the mean curvature of ∂Ω achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.