Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem

In this paper we study the shape of least-energy solutions to the quasilinear problem ε^m Δ_m u - u^{m - 1} + f (u) = 0 with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that as ε → 0⁺, the global maximum point P_ε 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.