Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system
Gustavo de Paula Ramos, Gaetano Siciliano
TL;DR
The paper studies semiclassical bound states for a quasilinear Schrödinger–Poisson system in $\mathbb{R}^3$ with a nonlinear term $u|u|^{p-1}$ and a higher-order electrostatic perturbation, in the regime $\varepsilon\to0$. It develops a variational Lyapunov–Schmidt framework: removing the Poisson component via a map $\phi_\varepsilon(u)$, building a manifold of approximate states $Z_\varepsilon$ around a ground-state profile $U$, and reducing to a finite-dimensional problem on a manifold $M$ where the external potential $V$ has a compact nondegenerate critical set. The authors obtain a precise expansion for the reduced energy $\widetilde J_\varepsilon$ showing its leading term scales like $V(\varepsilon z)^\theta$ plus controlled remainders, and prove that the reduced problem has at least cupl$(M)+1$ critical points for small $\varepsilon$, yielding multiple semiclassical states that concentrate near $M$. In the singleton case, the results imply the existence and concentration of solutions near a nondegenerate critical point of $V$. The work extends Lyapunov–Schmidt techniques to a quasilinear Schrödinger–Poisson system with a higher-order perturbation and links solution multiplicity to the topology of the potential landscape via cup-length.
Abstract
In the paper we consider the following quasilinear Schrödinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 Δu + (V + φ) u = u |u|^{p - 1} \newline - Δφ- βΔ_4 φ= u^2, \end{cases} $$ where $1 < p < 5, β> 0,V :\mathbb R^{3}\to ]0, \infty[$ and look for solutions $u,φ:\mathbb R^{3}\to \mathbb R$ 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 $V$.