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 is an element of 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 integral R3|del u|2dy Delta u. 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.