Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity

Abstract

In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity \begin{equation*} \begin{cases} -\Delta u+\phi u= |u|^{p-2}u\log|u|+\frac{\lambda}{u^\gamma}, &\rm \mathrm{in}\ \Omega, \\ -\Delta \phi=u^{2}, &\rm \mathrm{in}\ \Omega, \\ u=\phi=0, &\rm \mathrm{on}\ \partial\Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^3$ is a smooth bounded domain with boundary $\partial\Omega$, $00$ is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

Keywords

• logarithmic schrödinger–poisson system
• multiplicity
• singularity
• positive solutions