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

We study the shape of least-energy solutions to the quasilinear elliptic equation ∊^mΔmu − u^m−1 + f(u) = 0 with homogeneous Neumann boundary condition as ∊ → 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 ∊ → 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∊ and dist(P∊, ∂Ω)/∊ goes to zero as ∊ → 0⁺. We also give an approximation result and find that as ∊ → 0⁺ the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O(∊) of P∊ where they concentrate.