Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 66, pp. 1-18.

Title: Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations
       
Authors: Lorena Soriano Hernandez (Univ. de Brasilia, Brazil)
         Gaetano Siciliano (Univ. de Sao Paulo, Brazil)
         
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.