A perturbation approach for the Schrödinger-Born-Infeld system: solutions in the subcritical and critical case

2020/11/19 by Gaetano Siciliano, Siciliano, Gaetano, Zhisu Liu +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2011.09793

openalex publication_date 2020/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the following Schrödinger-Born-infeld system with a general nonlinearity \ -\triangle u+u+ϕu=f(u)+μ|u|⁴u · amp;in \R³,
-\textrmdiv((∇ϕ)/(√(1-|∇ϕ|²)))=u² · amp;in \R³,
u(x)→0, ϕ(x)→0, · amp; as x→∞, . where μ≥0 and f∈ C(\R,\R) satisfies suitable assumptions. This system arises from a suitable coupling of the nonlinear Schrödinger equation and the Born-Infeld theory. We use a new perturbation approach to prove the existence and multiplicity of nontrivial solutions of the above system in the subcritical and critical case. We emphasise that our results cover the case f(u)=|u|^p-1u for p∈(2,5/2] and μ=0 which was left in \citeAzzollini19 as an open problem.

Citations

• Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations
• A relation between pointwise convergence of functions and convergence of functionals
• A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity

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