Concentration behavior and local uniqueness of normalized solutions for Kirchhoff type equation

Abstract

   Let $a>0$, $b>0$ and $V(x)\ge 0$ be a coercive function in $\mathbb R^2$. We study the solutions with normalized $L^2$-norm for the following Kirchhoff type equation

$$\begin{aligned} -\left( a+b\int \limits _{\mathbb R^2}|\nabla u|^2\hbox {d}x\right) \Delta u+V(x)u=\beta |u|^{2}u+\lambda u \end{aligned}$$

on a suitable weighted Sobolev space

$$\begin{aligned} \mathcal {H}=\left\{ u\in H^{1}(\mathbb R^2):\int \limits _{\mathbb R^2}V(x)u^2\textrm{d}x<\infty \right\} . \end{aligned}$$

Our aim is to investigate the limit behaviors of the solutions with normalized $L^2$-norm for this equation as $(a,b)\rightarrow (0,0)$. Moreover, the uniqueness of the solution with normalized $L^2$-norm for this equation is also discussed for a, b close to 0