Concentration of Normalized Solutions for Mass Supercritical Kirchhoff Type Equations

In this paper, we study the existence and concentration behavior of normalized positive solutions to the following \(L^2\) $L^2$ -supercritical Kirchhoff type equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\left( a\varepsilon ^2+b\varepsilon \displaystyle \int _{\mathbb {R}^3}|\nabla v|^2dx\right) \Delta v+\lambda v=K(x)|v|^{p-2}v, \,\, x\in \mathbb {R}^3, \\ \displaystyle \int _{\mathbb {R}^3}|v|^2dx=m\varepsilon ^3,\,\,v\in H^1(\mathbb {R}^3),\\ \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\left({\displaystyle a{\epsilon }^2+b\epsilon {\int }_{{\mathbb{R}}^3}{|\mathrm{\nabla }v|}^{2}dx}\right)\mathrm{\Delta }v+\lambda v=K(x){|v|}^{p-2}v,x\in {\mathbb{R}}^3,\hfill \\ {\displaystyle {\int }_{{\mathbb{R}}^3}{|v|}^{2}dx=m{\epsilon }^3,v\in H^1({\mathbb{R}}^3),}\hfill \end{array}\right)\end{array}$$ where \(a, b, m>0\) $a,b,m>0$ , \(p\in \left( \frac{14}{3},6\right) \) $p\in \left(\frac{14}{3},,,6\right)$ , \(\varepsilon \) $\epsilon$ is a small positive parameter, K is a positive continuous function possessing a local maximum point and \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ will arise as a Lagrange multiplier. We construct a family of positive normalized solutions \(v_\varepsilon \in H^1(\mathbb {R}^3)\) $v_{\epsilon }\in H^1({\mathbb{R}}^3)$ which concentrates around the local maximum of K as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ by using a combination of the variational approach and a penalization technique. Moreover, we also give the expression of the approximation solutions.