Morse index for solutions of the nonlinear Schr?dinger equation in a degenerate setting

摘要

In this paper, we study the Morse index of the single-peak solutions concentrating at a point P? ? N of the problem $$ {mathbb p}_{epsilon} left{begin{array}{l} -epsilon^2 {rm Delta} u + V(x) u= u^p mbox{in} {mathbb R}^N[3pt] u>0 mbox{in} {mathbb R}^N[3pt] u in H^1({mathbb r}^n), end{array}right. $$ where ε is a positive parameter, $1≤ V(x)≤ V 1. The point P is a critical point of V (possibly degenerate). We will also consider some ‘strong degeneracy’ of P like the case where P belongs to a flat region of maxima.