Electron. J. Differential Equations, Vol. 2023 (2023), No. 66, pp. 1-18.

Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations

Lorena Soriano Hernandez, Gaetano Siciliano

We study the existence and multiplicity of solutions for the Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u + \phi u = \omega u \quad\text{ in } \Omega \cr a^2\Delta^2\phi-\Delta \phi = u^2 \quad\text{ in } \Omega \cr u=\phi=\Delta\phi=0\quad\text{ on } \partial\Omega \cr \int_{\Omega} u^2\,dx =1 }$$ where \(\Omega\) is an open bounded and smooth domain in \(\mathbb R^{3}\), \(a>0 \) is the Bopp-Podolsky parameter. The unknowns are \(u,\phi:\Omega\to \mathbb R\) and \(\omega\in\mathbb R\). By using variational methods we show that for any \(a>0\) there are infinitely many solutions with diverging energy and divergent in norm. We show that ground states solutions converge to a ground state solution of the related classical Schrodinger-Poisson system, as \(a\to 0\).

Submitted March 8, 2023. Published October 13, 2023.
Math Subject Classifications: 35A15, 58E05.
Key Words: Schrodinger type systems; existence of solutions; variational methods; critical point theory.
DOI: 10.58997/ejde.2023.66

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Lorena Soriano Hernandez
Departamento de Matemática
Universidade de Brasília - UnB
70910-900, Campus Universitário Darcy Ribeiro
Asa Norte, Brasília, DF, Brazil
email: loresohe199@gmail.com
Gaetano Siciliano
Departamento de Matemática
Universidade de São Paulo
Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil
email: sicilian@ime.usp.br

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