On multi-bump solutions for a class of Schrödinger-Poisson systems with $p$-Laplacian in ${\mathbb{R}}^3$

In this article, we consider the following a class of Schrödinger-Poisson systems with $p$-Laplacian in ${\mathbb{R}}^3$ of the form:

$$\{\begin{array}{ll}-{\mathrm{\Delta }}_{p}u+(\lambda b(x)+1)|u{|}^{p-2}u+\varphi |u{|}^{s-2}u=g(u)& \text{in}{\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =|u{|}^s& \text{in}{\mathbb{R}}^3,\end{array}$$

where $1<p<3$, $\frac{p}{2}<s<p$, ${\mathrm{\Delta }}_{p}u:=div(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)$ is the $p$-Laplacian operator, $\lambda$ is a positive parameter. Assume that the nonnegative function $b$ possesses a potential well int $(b^{-1}(\{0\}))$, which is composed of $k$ disjoint components ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2,\cdots ,{\mathrm{\Omega }}_k$ and consider the nonlinearity $g$ with subcritical growth. Using the variational methods and Morse iteration technique, the existence of positive multi-bump solutions are obtained.