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

2025/02/18 by Edwin G. Murcia, Murcia, Edwin G., Gaetano Siciliano +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2502.12626

openalex publication_date 2025/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a smooth bounded domain Ω⊂ \mathbb R³, we consider the following nonlinear Schrödinger-Poisson type system \ -Δu+ ϕu -\absu^p-2u = ωu · amp; in λΩ, -Δϕ=u² · amp; in λΩ, u · gt;0 · amp; in λΩ, u =ϕ=0 · amp; on ∂ (λΩ), ∫_λΩu² d x=ρ² . in the expanding domain λΩ⊂ \mathbb R³, λ>1 and p∈ (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→+∞ the solutions found converge to a ground state of the problem in the whole space \mathbb R³.

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• Ground state solutions for the asymptotically periodic Schrödinger-Poisson systems with p-Laplacian

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