On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems

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We study the shape of least-energy solutions to the quasilinear elliptic equation εmΔmuum−1 + f(u) = 0 with homogeneous Neumann boundary condition as ε → 0+ in a smooth bounded domain Ω ⊂ ℝN. 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 Pεand dist(Pε, ∂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 Pεwhere they concentrate..