Morse index of concentrated solutions for the nonlinear Schrödinger equation with a very degenerate potential

We revisit the following nonlinear Schrödinger equation

$$\begin{aligned} -\varepsilon ^2\Delta u+ V(x)u=u^{p},\quad u>0,\;\; u\in H^1({\mathbb {R}}^N), \end{aligned}$$

where \(\varepsilon >0\) is a small parameter, \(N\ge 2\) and \(1<p<2^*-1\). It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive k-peak solutions to above problem when the critical points of V(x) are non-isolated and degenerate. We also give a specific formula for the Morse index of k-peak solutions when the critical point set of V(x) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential V(x). Our results generalize Grossi and Servadei’s work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions.