Lorena Soriano Hernandez, Gaetano Siciliano
Abstract:
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 |
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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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