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

摘要

We study the shape of least-energy solutions to the quasilinear elliptic equation epsilon(m)Delta(m)u - u(m-1) + f(u) = 0 with homogeneous Neumann boundary condition as epsilon -> 0(+) in a smooth bounded domain Omega subset of < Ropf >(N). Firstly, we give a sharp upper bound for the energy of the least-energy solutions as epsilon -> 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-epsilon and dist(P-epsilon, partial derivative Omega)/epsilon goes to zero as epsilon -> 0(+). We also give an approximation result and find that as epsilon -> 0(+) the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O(epsilon) of P-epsilon where they concentrate.