Solutions for Mass Subcritical and Supercritical Schrödinger-Bopp-Podolsky Type System With Logarithmic Nonlinearity

In this In this paper, we study existence of normalized solutions for the following Schr\"{o}dinger-Bopp-Podolsky type system involving a logarithmic nonlinearity \[ \left\{ \begin{array}{ll} \displaystyle -\Delta_p u-\phi u=\lambda |u|^{p-2}u+|u|^{p-2}u\log|u|^p +\mu|u|^{q-2}u & \ \ \text{in}\ \mathbb{R}^3, \medskip \\ \displaystyle -\Delta\phi+a^2\Delta^2\phi=4\pi u^2 & \ \ \text{in}\ \mathbb{R}^3, \medskip\\ \displaystyle \int_{\R^3}|u|^pdx=d^p, \end{array} \right. \] where $\Delta_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ is the usual $p-$Laplacian operator, $q\in (p, p^{*})$ and $d,a,\mu>0$ are parameters. The unknowns are $u,\phi:\mathbb R^{3}\to \mathbb R$ and the Lagrange multiplier $\lambda\in\R$. We show that if $p\in[2,\frac{12}{5})$, there is a solution in both mass-subcritical and mass-supercritical cases. We show also that the solutions found converge to a solution of the related Schr\"odinger-Poisson type system as $a\to 0$.