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

Abstract

In this note, we consider the problem

$$-\Delta u = u^p \quad {\rm{in}}\; \Omega, \quad u > 0 \quad {\rm{in}} \; \Omega, \quad u |_{\partial \Omega} = 0$$

on a smooth bounded domain Ω in ${\mathbb {R}^2}$ for p > 1. Let u _p be a positive solution of the above problem with Morse index less than or equal to ${m \in \mathbb {N}}$ . We prove that if u _p further satisfies the assumption ${p {\int_\Omega} |\nabla u_p|^2 dx = O(1)}$ as p → ∞, then the number of maximum points of u _p is less than or equal to m for p sufficiently large. If Ω 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.