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 εmΔmu − um−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..
& Zhao, C.
(2007). On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems. IMA Journal of Applied Mathematics, 72 (2), 113-139.