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

Abstract

A non-degeneracy theorem for positive solutions of the standard Kirchhoff model is proved in all cases except for n ≥ 5 and b ∫ R n | ∇ Q | 2 d x = 2 ( n − 4 ) n − 4 2 / ( n − 2 ) n − 2 2 . In particular, for dimensions n ≥ 5 and b ∫ R n | ∇ Q | 2 d x < 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 Schrödinger 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.