Abstract:

In this paper, we obtain the local uniqueness of a single peak solution to the following Kirchhoff problem

$-\Big(\epsilon^2 a+ \epsilon b \int_{\mathbb{R} ^3} |\nabla u|^2{\rm d}x\Big) \Delta u + u=K(x)|u|^{p-1}u, u>0,x \in \mathbb{R} ^3,$

for ε>0 sufficiently small, where a, b>0 and 1 < p < 5 are constants, K: $\mathbb{R}^3$→$\mathbb{R}$ isabounded continuous function. We mainly use a contradiction argument developed by Li G, Luo P, Peng S in[20], applying some local pohozaev identities. Our result is totally new for Kirchhoff equations.

Key words: Local uniqueness, Nonlinear Kirchhoff equations, Pohozaev identities