Existence and concentration of semiclassical bound states for a quasilinear Schr\"odinger-Poisson system

TL;DR

This paper investigates the existence and concentration behavior of semiclassical bound states for a quasilinear Schrödinger-Poisson system in three-dimensional space, employing Lyapunov-Schmidt reduction to relate solutions to the topology of the potential.

Contribution

It introduces a novel application of Lyapunov-Schmidt reduction to estimate the number of solutions based on the cup-length of the potential's critical manifold.

Findings

Established existence of solutions in the semiclassical limit

Linked solution multiplicity to topological properties of the potential

Provided estimates for the number of bound states

Abstract

In the paper we consider the following quasilinear Schr\"odinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p - 1} \newline - \Delta \phi - \beta \Delta_4 \phi = u^2, \end{cases} $$ where $1 < p < 5, \beta > 0,V :\mathbb R^{3}\to ]0, \infty[$ and look for solutions $u,\phi:\mathbb R^{3}\to \mathbb R$ in the semiclassical regime, namely when $\varepsilon\to 0.$ 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 $V$.