Locating the Peaks of Least-Energy Solutions to a Quasi-Linear Elliptic Neumann Problem, Part II

We continue our work (Y. Li, C. Zhao, Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, J. Math. Anal. Appl. 336 (2007) 1368-1383) to 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. In this paper we focus on the case 1 < m < 2 as a complement to our previous work on the case m ≥ 2. We use an intrinsic variation method to show that as the case m ≥ 2, when ε → 0⁺, the global maximum point P_ε of least-energy solutions goes to a point on the boundary ∂ Ω at a rate of o (ε) and this point on the boundary approaches a global maximum point of mean curvature of ∂ Ω.