Local uniqueness of multi-peak solutions to a class of Kirchhoff equations

Abstract

where ǫ > 0 is a parameter, V : R → R is a bounded continuous function. Under some mild conditions on V , Luo, Peng, Wang and the last named author of the present paper [22] proved the existence of multi-peak solutions to (1.1). As a continuation of the work [22], this paper is devoted to establish a local uniqueness result for the multi-peak solutions obtained there. For physical background for equation (1.1), the readers are referred to Luo et al. [22] and the references therein. To be precise, we first give the definition of k-peak solutions of Eq. (1.1) as usual. Definition 1.1. Let k ∈ N, bj ∈ R , 1 ≤ j ≤ k. We say that uǫ ∈ H (R) is a k-peak solution of (1.1) concentrated at {b1, b2, · · · , bk}, if (i) uǫ has k local maximum points x j ǫ ∈ R , j = 1, 2, . . . , k, satisfying xǫ → bj as ǫ→ 0 for each j; (ii) For any given τ > 0, there exists R ≫ 1, such that |uǫ(x)| ≤ τ for x ∈ R \ ∪j=1 BRǫ(x j ǫ);