Morse index for solutions of a nonlinear Kirchhoff equation

In this paper, we study a perturbed nonlinear Kirchhoff equation with subcritical growth in R3. Although the existence of concentrated solutions with a single peak or multi peaks to the problem above has been obtained in Li et al. [J. Differ. Equations 268, 541–589 (2020)] and Luo et al. [Proc. R. Soc. Edinburgh, Sect. A 149, 1097–1122 (2019)], respectively, the Morse indices of them remain open. First, we compute the Morse index of single-peak solutions concentrated at a point P∈R3 by variational methods, which can also be applied to the case where P is a degenerate critical point of V. Then, we also study the Morse index of multi-peak solutions concentrated at the non-degenerate critical points of V. Here the main difficulty comes from the nonlocal term ∫R3|∇u|2dyΔu. In addition, since the estimates of the eigenvalues and the eigenfunctions of the linearized problem associated to the limit problem are unknown, we have to study them independently, which are quite interesting.