On the shape of least-energy solutions for a class of quasilinear elliptic Neumann problems

Abstract

We study the shape of least-energy solutions to the quasilinear elliptic equation ε^mΔ_mu − u^m−1 + f(u) = 0 with homogeneous Neumann boundary condition as ε → 0⁺ in a smooth bounded domain Ω ⊂ ℝ^N. Firstly, we give a sharp upper bound for the energy of the least-energy solutions as ε → 0⁺, which plays an important role to locate the global maximum. Secondly, based on this sharp upper bound for the least energy, we show that the least-energy solutions concentrate on a point P_ε and dist(P_ε, ∂Ω)/ε goes to zero as ε → 0⁺. We also give an approximation result and find that as ε → 0⁺ the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O(ε) of P_ε where they concentrate.