On the critical Schrödinger-Poisson system with $ p $-Laplacian

Abstract

<p style='text-indent:20px;'>In this paper we consider the critical quasilinear Schrödinger-Poisson system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Delta_p u+|u|^{p-2}u+\mu\phi |u|^{p-2}u = \lambda|u|^{q-2}u+|u|^{p^*-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ -\Delta \phi = |u|^p, &\mathrm{in}\ \mathbb{R}^3, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \frac{3}{2}<p<3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \Delta_p u = \hbox{div}(|\nabla u|^{p-2}\nabla u) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ p<q<p^*: = \frac{3p}{3-p} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mu,\lambda>0 $\end{document}</tex-math></inline-formula>. Based upon the variational approach, the ground state solutions and the nontrivial solutions are obtained depending on the parameters <inline-formula><tex-math id="M6">\begin{document}$ q $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>.</p>

Topics & Concepts