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 a singularly perturbed quasilinear problem with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that at limit, the global maximum point of least-energy solutions goes to a point on the boundary faster than the linear rate and this point on the boundary approaches to a point where the mean curvature of the boundary achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.

Keywords: Quasilinear Neumann problem, m-Laplacian operator, least-energy solution, exponential decay, mean curvature.