Existence and Concentration of Semiclassical Bound States for a Quasilinear Schrödinger-Poisson System

In the paper we consider the following quasilinear Schrödinger–Poisson system in the whole space \({\mathbb {R}}^{3}\) ${\mathbb{R}}^3$ \(\begin{aligned} {\left\{ \begin{array}{ll} - \varepsilon ^2 \Delta u + ({V + \phi }) u = u \left| {u}\right| ^{p - 1} \\ - \Delta \phi - \beta \Delta _4 \phi = u^2, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\epsilon }^2\mathrm{\Delta }u+(V+\varphi )u=u{\left|u\right|}^{p-1}\hfill \\ -\mathrm{\Delta }\varphi -\beta {\mathrm{\Delta }}_4\varphi =u^2,\hfill \end{array}\right)\end{array}$$ where \(1< p < 5, \beta > 0,V:{\mathbb {R}}^{3}\rightarrow ]0, \infty [\) $1<p<5,\beta >0,V:{\mathbb{R}}^3\to ]0,\infty [$ , and look for solutions \(u,\phi :{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\) $u,\varphi :{\mathbb{R}}^3\to \mathbb{R}$ in the semiclassical regime, namely when \(\varepsilon \rightarrow 0.\) $\epsilon \to 0.$ By means of the Lyapunov–Schmidt method we estimate the number of solutions by the cup-length of the critical manifold of the external potential V.