Small normalised solutions for a Schrödinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour

TL;DR

This work analyzes a Schrödinger–Poisson system on expanding domains $λΩ$ with fixed mass $ρ$ and subcritical nonlinearity $p∈(2,3)$. Using a constrained variational approach and the barycenter technique, it proves at least ${ m cat}(m{Ω})$ positive solutions for sufficiently large $λ$, with negative Lagrange multipliers, and shows these solutions converge (up to translations) to the ground state of the limit problem in $m{R}^3$ as $λ→∞$. The method foregrounds domain topology via Ljusternik–Schnirelmann category and handles the nonlocal Poisson term, including a radial reduction to obtain compactness. An appendix clarifies the essential distinctions between bounded-domain and whole-space minimization, especially the sign of the constrained minimum and its behavior under domain expansion.

Abstract

Given a smooth bounded domain $Ω\subset \mathbb R^3$, we consider the following nonlinear Schrödinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -Δu+ φu -\abs{u}^{p-2}u = ωu & \quad \text{in } λΩ, -Δφ=u^{2}& \quad \text{in }λΩ, u>0 &\quad \text{in }λΩ, u =φ=0 &\quad \text{on }\partial (λΩ), \int_{λΩ}u^{2} \,\text{d} x=ρ^2 \end{array} \right. \end{equation*} in the expanding domain $λΩ\subset \mathbb R^{3}, λ>1$ and $p\in (2,3)$, in the unknowns $(u,φ,ω)$. We show that, for arbitrary large values of the expanding parameter $λ$ and arbitrary small values of the mass $ρ>0$, the number of solutions is at least the Ljusternick-Schnirelmann category of $λΩ$. Moreover we show that as $λ\to+\infty$ the solutions found converge to a ground state of the problem in the whole space $\mathbb R^{3}$.

Theorems & Definitions (8)

• Theorem 1.1
• Remark 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Lemma 5.1
• Proposition 5.2