Electron. J. Differential Equations, Vol. 2023 (2023), No. 66, pp. 118.
Existence and asymptotic behavior of solutions to eigenvalue problems for SchrodingerBoppPodolsky equations
Lorena Soriano Hernandez, Gaetano Siciliano
Abstract:
We study the existence and multiplicity of solutions for the
SchrodingerBoppPodolsky 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 BoppPodolsky 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 SchrodingerPoisson 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
Show me the PDF file (371 KB),
TEX file for this article.

Lorena Soriano Hernandez
Departamento de Matemática
Universidade de Brasília  UnB
70910900, 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, 05508090 São Paulo, SP, Brazil
email: sicilian@ime.usp.br

Return to the EJDE web page