Existence and concentration of normalized solutions for logarithmic Schrödinger–Bopp–Podolsky type system

In this paper, we are interested in the existence and concentration of normalized solutions for the following logarithmic Schrödinger–Bopp–Podolsky type system involving the $p$ -Laplacian in $\mathbb{R}^3$ : \begin{equation*}\left\{\begin{array}{ll}\displaystyle -\varepsilon^p\Delta_p u+Z(x)|u|^{p-2}u-\kappa\phi u=\lambda |u|^{p-2}u+|u|^{p-2}u\log|u|^p& \text{in} \ \mathbb{R}^3, \\\displaystyle -\varepsilon^2\Delta\phi+a^2\varepsilon^4\Delta^2\phi=4\pi u^2& \text{in} \ \mathbb{R}^3, \\\displaystyle \int_{\mathbb{R}^3}|u|^pdx=d^p\varepsilon^3,\end{array}\right.\end{equation*} where $\Delta_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$ -Laplacian operator, $Z$ is a given external potential, $\kappa \gt 0$ a constant, $a \gt 0$ is the Bopp–Podolsky constant and $\varepsilon \gt 0$ is a small parameter. The unknowns are $u,\phi:\mathbb{R}^{3}\to \mathbb{R}$ and the Lagrange multiplier $\lambda\in\mathbb{R}$ . If $p\in[2,\frac{12}{5})$ , we obtain, via the variational method, that the number of positive solutions depends on the profile of $Z$ and the solutions concentrate around the global minimum points of $Z$ in the semiclassical limit as $\varepsilon\to 0^{+}$ .