Normalized Solutions for Schrödinger-Bopp-Podolsky Systems in Bounded Domains with General Nonlinearities

2025/11/26 by Sheng, Kai
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.21303

Abstract

In this paper, by adapting the perturbation method, we study normalized standing wave solutions for the following nonlinear Schrödinger-Bopp-Podolsky system: - Delta u + q(x) phi u = omega u + f(u) in Omega, - Delta phi + a² Delta² phi = q(x) u² in Omega, where Omega is a smooth bounded domain in R³, a > 0, and omega is the Lagrange multiplier associated with the L² mass constraint integral over Omega of u² equals mu, and f: R -> R is a continuous function satisfying some technical conditions. In particular, we prove the existence of normalized solutions for all masses mu in an interval (0, mu₀), under either Navier or Neumann boundary conditions for phi. Moreover, when f is odd, we obtain multiplicity of normalized solutions; and if Omega is star-shaped, we further obtain a normalized ground state solution.

Citations

• Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions
• On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence
• Normalized solutions for Schrödinger-Bopp-Podolsky systems in bounded domains
• Multiple solutions for a Schrödinger-Bopp-Podolsky system with positive potentials
• Detailed proof of classical Gagliardo-Nirenberg interpolation inequality\n with historical remarks
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• Multipole Expansion in Generalized Electrodynamics
• Normalized bound states for the nonlinear Schrodinger equation in\n bounded domains
• Eine lineare Theorie des Elektrons

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