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 ${\mathbb{R}}^3$ $$\{\begin{array}{l}-{\epsilon }^2\mathrm{\Delta }u+(V+\phi )u=u\mathrm{\mid }u{\mathrm{\mid }}^{p-1}\\ -\mathrm{\Delta }\phi -\beta {\mathrm{\Delta }}_4\phi =u^2,\end{array}$$ where $1<p<5,\beta >0,V:{\mathbb{R}}^3\to ]0,\mathrm{\infty }[$ and look for solutions $u,\phi :{\mathbb{R}}^3\to \mathbb{R}$ in the semiclassical regime, namely when $\epsilon \to 0.$ 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 $V$.