Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation

2021/03/31 by Louis Jeanjean, Thanh Trung Le · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Combinatorics #Critical point (mathematics) #Energy (signal processing) #Lambda #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Norm (philosophy) #Physics #Poisson's equation #Quantum mechanics #Schrödinger equation #Sobolev space #math.AP

paper · pdf · doi:10.1016/j.jde.2021.09.022

published in Journal of Differential Equations 303, 277-325 (Elsevier BV) · This version is the final one, corresponding to the paper now published in Journal of Differential Equations

openalex publication_date 2021/09/28 · arxiv created 2021/10/09 · arxiv updated 2021/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We look for solutions to the Schrödinger-Poisson-Slater equation - Δu + λu - γ(|x|^-1 * |u|²) u - a |u|^p-2u = 0 in ℝ³, which satisfy ∫_ℝ³|u|² dx = c for some prescribed c>0. Here u ∈ H¹(ℝ³), γ∈ ℝ, a ∈ ℝ and p ∈ ((10)/(3), 6]. When γ>0 and a > 0, both in the Sobolev subcritical case p ∈ ((10)/(3), 6) and in the Sobolev critical case p=6, we show that there exists a c₁>0 such that, for any c ∈ (0,c₁), the equation admits two solutions u_c⁺ and u_c^- which can be characterized respectively as a local minima and as a mountain pass critical point of the associated \it Energy functional restricted to the norm constraint. In the case γ>0 and a < 0, we show that, for any p ∈ ((10)/(3),6] and any c>0, the equation admits a solution which is a global minimizer. Finally, in the case γ<0, a >0 and p=6 we show that it does not admit positive solutions.

Citations

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