Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in ${\mathbb{R}}^3$

Abstract

In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems that there have no this type of results even on positive ground states solutions to Kirchhoff type equations, much less on general positive solutions. To overcome the difficulty brought by the nonlocality, some new observation on Kirchhoff equations is found, and some related theories on classical Schrödinger equations are applied.

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Cited by in corpus (2)

• Existence and multiplicity of bound state solutions to a Kirchhoff type equation with a general nonlinearity
• Concentration of positive ground state solutions for critical Kirchhoff equation with competing potentials