Chain-structure solutions to a Schrödinger-Poisson system in $\mathbb{R}^3$

TL;DR

This work addresses chain-structure (periodic in the third coordinate) solutions to the Schrödinger–Poisson system in $\mathbb{R}^3$ by first constructing a Green function for the periodic Poisson problem on a slab $\Omega = \mathbb{R}^2 \times (-\ell,\ell)$ and analyzing its asymptotics, which interpolate between 2D logarithmic and 3D Newtonian behaviors. A variational Choquard framework is then developed on a log-weighted space $X$, with kernel decomposition $K=K_1+K_2$, enabling the definition of a well-posed energy functional $\Phi$ and a Nehari manifold, together with symmetry reductions to obtain nontrivial $x_3$-variations. Using Ljusternik–Schnirelmann theory in the radially symmetric setting, the authors prove the existence of a sequence of radial ground states and high-energy solutions, including regularity of periodic extensions. In the constant-$a$ along $x_3$ case, they show radial ground states are fully nontrivial for large slab width $\ell$, and they extend this to all $\ell>0$ via a $G$-invariant symmetric criticality framework, yielding infinitely many fully nontrivial chain-structure solutions. The results bridge planar and volumetric Poisson interactions and provide a robust variational approach to nonlocal Schrödinger–Poisson systems with periodic structure.

Abstract

We prove the existence of ground states and high-energy solutions to the following Schrödinger-Poisson system \begin{align*} \begin{cases} - Δu + a(x) u + u v = 0,\newline Δv = u^2, \end{cases} \quad \text{in } \mathbb{R}^3, \end{align*} where $a \in L^\infty(\mathbb{R}^3)$ is nonnegative and radially symmetric in the first two variables. Differing from the standard approach, our framework yields chain-structure solutions, i.e. solutions periodic in the third variable. A central part of this work is the construction of the Green function of a Poisson problem subject to periodic boundary conditions and we show that its asymptotic profile is tightly related to both the two and three dimensional Poisson problems in the entire space. If the potential $a$ is constant along the third variable, we apply symmetry techniques to construct solutions that have nonvanishing derivative in the third variable.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm:asymtoticsGreen']} (Item 1)
• Corollary 2.2
• Remark 2.1