Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system

Abstract

In the paper we consider the following quasilinear Schrödinger--Poisson system in the whole space ${\mathbb{R}}^3$ $$\{\begin{array}{l}-{\epsilon }^2\Delta u+(V+\varphi )u=u|u{|}^{p-1}\\ -\Delta \varphi -\beta {\Delta }_4\varphi =u^2,\end{array}$$ where $1<p<5,\beta >0,V:{\mathbb{R}}^3\to ]0,\infty [$ and look for solutions $u,\varphi :{\mathbb{R}}^3\to \mathbb{R}$ 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 $V$.

19 pages; comments are welcome