Non-degeneracy of positive solutions of Kirchhoff equations and its application

A non-degeneracy theorem for positive solutions of the standard Kirchhoff model is proved in all cases except for n >= 5 and b integral(n)(R) vertical bar del Q vertical bar(2)dx = 2(n - 4) (n -4/2)/(n - 2) (n-2/2). In particular, for dimensions n >= 5 and b integral(n)(R) vertical bar del Q vertical bar(2)dx = 2(n - 4)(n -4/2)/n - 2) (n-2/2), we show that there exist two non -degenerate positive solutions of the Kirchhoff equations, which seem to be completely different from the result of the standard Schrodinger equation. The effects of the non-local term on the positive solution set are also studied. We show that the non-local term has no effect on the structure of the positive solution set for dimensions 1 <= n <= 3 and the effects eventually appear for dimensions n >= 4. With the non-degeneracy property of positive solutions of the limit problem, we construct concentrated solutions of a singularly perturbed Kirchhoff problem via the well-known Lyapunov Schmidt reduction method. For dimensions n >= 5, the existence result of concentrated solutions is completely new and is not implied by the main result of G.M. Figueiredo et al. in [19], where a generalized singularly perturbed Kirchhoff problem was considered. (C) 2018 Elsevier Inc. All rights reserved.