Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem

摘要

In this note, we consider the problem -Δu = u~P in _, u > 0 in _ , u|__ = 0 on a smooth bounded domain _ in R~2 for p > 1. Let u_p be a positive solu_tion of the above problem with Morse index less than or equal to m ∈ N. We prove that if u_p further satisfies the assumption p ∫_ |_u_p|~2 dx = O(1) as p→_, then the number of maximum points of u_p is less than or equal to m for p sufficiently large. If _ is convex, we also show that a solution of Morse index one satisfying the above assumption has a unique critical point and the level sets are star-shaped for p sufficiently large.