Multiple Solutions for Logarithmic Schrödinger–Poisson Systems with a Small Perturbation

We consider the following logarithmic Schrödinger–Poisson system: \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\phi u=\left| u \right| ^{p-2}u \ln u^{2}+\lambda f(x,u),& \hbox {in}\ \Omega ,\\ -\Delta \phi =u^{2},& \hbox {in}\ \Omega ,\\ \phi ,u=0,& \hbox {on}\ \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+\varphi u={\left|u\right|}^{p-2}uln{u}^2+\lambda f(x,u),\hfill & \text{in}\mathrm{\Omega },\hfill \\ -\mathrm{\Delta }\varphi =u^2,\hfill & \text{in}\mathrm{\Omega },\hfill \\ \varphi ,u=0,\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded domain in \({\mathbb {R}}^{3}\) ${\mathbb{R}}^3$ with smooth boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ , \(p\in (4,6)\) $p\in (4,6)$ , f(x, u) is continuous without any other condition. Using constrained variational method, Mountain Pass Theorem and iterative technique, we prove the existence of mountain pass solutions when \(\lambda >0\) $\lambda >0$ small enough. Moreover, with \(f(x,0)\ne 0\) $f(x,0)\ne 0$ in \(\Omega \) $\mathrm{\Omega }$ , the above system possesses the another local minimum nontrivial solution. Finally, we prove that for any \(j\in {\mathbb {N}}\) $j\in \mathbb{N}$ , there exists \(\lambda _{j}>0\) ${\lambda }_j>0$ , such that if \(0<\lambda <\lambda _{j}\) $0<\lambda <{\lambda }_j$ , the above system possesses at least j distinct high energy solutions.