Uniqueness and Nondegeneracy of Positive Solutions of General Kirchhoff Type Equations

Abstract

In the present paper, we study uniqueness and nondegeneracy of positive solutions to the general Kirchhoff type equations

$$-M\left(\int_{\mathbb{R}^{N}}{\vert\nabla v\vert}^{2}dx\right)\Delta v=g(v) \quad {\rm in}\;{\mathbb{R}^{N}},$$

where M: [0, +∞) ↦ ℝ is a continuous function satisfying some suitable conditions and v ∈ H¹(ℝ^N). Applying our results to the case M(t) = at + b, a, b > 0, we make it clear all the positive solutions for all dimensions N ≥ 1. Our results can be viewed as a generalization of the corresponding results of Li et al. [JDE, 2020, 268, Section 1.2].