Abstract:

In this paper we consider the existence of solutions to the nonhomogeneous quasilinear Kirchhoff- Schr\"{o}dinger-Poisson system with $ p $-Laplacian:

\begin{equation*}\label{1ip}

\begin{cases}\displaystyle

- \Big(a-b \int_{\mathbb{R}^3}|\nabla u|^p{\rm d}x \Big)\Delta_p u+|u|^{p-2}u+\lambda\phi_uu= |u|^{q-2}u+h(x), &\text{ }x\in \mathbb{R}^3,\\

-\Delta\phi=u^2, &\text{ }x\in \mathbb{R}^3,\\

\end{cases}

\end{equation*}

where $ a,b>0 $, $ \frac{4}{3} < p < \frac{12}{5} $, $ p < q < p^* =\frac{3p}{3-p} $, $ \lambda > 0 $. Under suitable assumptions on $ h(x) $, the existence of multiple solutions of the system is obtained by using the Ekeland variational principle and the Mountain pass theorem.

Key words: Kirchhoff-Schr\"odinger-Poisson system, $ p $-Laplacian, Ekeland variational principle, Mountain pass theorem