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>

The quasilinear Schrödinger--Poisson system

“…We also note that the existence results of this paper coincide with those in [14,15,21] in the case q = 2, s = 2 or s = p. In particular, we extend the results in [14,15] to q = 2.…”

“…To the best of our knowledge, this is the first time that (1.1) is considered in R 3 for q = 2 in the mathematical literature. We emphasize that the system (1.1) is a new model of variational methods and it generalizes the system studied in [14,15,21].…”

“…This is the main object of our paper. Motivated by the above mentioned works, especially our recent works [14,15], this paper deals with the case q = 2, or to be more precise, 1 < p < 3, q and s satisfy (1.2), see the figure F2. From the figure F2, we see that the case q = 2 is included in the present paper.…”

“…the coordinate point (2, 2, 2) in the figure F1. In [14] the system (1.1) has been considered for the case that 4 3 < p < 12 5 , q = 2 and s = 2, while in [15] the system (1.1) has been considered for the case that 1 < p < 3, q = 2 and s = p, see the blue and the red line segment in the figure F1, respectively.…”

Multiplicity and concentration of positive solutions for quasilinear Schrödinger-Poisson system with critical nonlinearity

“…To the best of our knowledge, the results of the existence of solution for system (1.1) are very few currently; we refer to [14,15]. Inspired by [14,15,18], in the paper we consider the multiplicity and concentration of solutions for system (1.1).…”