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

   Let \(a>0\) $a>0$ , \(b>0\) $b>0$ and \(V(x)\ge 0\) $V(x)\ge 0$ be a coercive function in \(\mathbb R^2\) ${\mathbb{R}}^2$ . We study the solutions with normalized \(L^2\) $L^2$ -norm for the following Kirchhoff type equation \(\begin{aligned} -\left( a+b\int \limits _{\mathbb R^2}|\nabla u|^2\hbox {d}x\right) \Delta u+V(x)u=\beta |u|^{2}u+\lambda u \end{aligned}\) $$\begin{array}{c}\hfill -\left(a,+,b,\underset{{\mathbb{R}}^2}{\int },{|\mathrm{\nabla }u|}^2,\text{d},x\right)\mathrm{\Delta }u+V(x)u=\beta {|u|}^{2}u+\lambda u\end{array}$$ on a suitable weighted Sobolev space \(\begin{aligned} \mathcal {H}=\left\{ u\in H^{1}(\mathbb R^2):\int \limits _{\mathbb R^2}V(x)u^2\textrm{d}x<\infty \right\} . \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{H}=\left\{u,\in ,H^1,({\mathbb{R}}^2),:,\underset{{\mathbb{R}}^2}{\int },V,(x),u^2,\text{d},x,<,\infty \right\}.\end{array}$$ Our aim is to investigate the limit behaviors of the solutions with normalized \(L^2\) $L^2$ -norm for this equation as \((a,b)\rightarrow (0,0)\) $(a,b)\to (0,0)$ . Moreover, the uniqueness of the solution with normalized \(L^2\) $L^2$ -norm for this equation is also discussed for a, b close to 0