Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations
DOI:
https://doi.org/10.58997/ejde.2023.66Keywords:
Schrodinger type systems; existence of solutions; variational methods; critical point theoryAbstract
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 \(a\to 0\).
For more information see https://ejde.math.txstate.edu/Volumes/2023/66/abstr.html
References
R. A. Adams, J. Fournier; Sobolev Spaces, Elsevier, Academic Press, 2003.
D. G. Afonso, G. Siciliano; Normalized solutions to a Schrodinger-Bopp-Podolsky system under Neumann boundary conditions, Commun. Contemp. Math., 25 No. 2 (2023) 2150100. 20 pages.
V. Benci, D. Fortunato; An Eigenvalue Problem for the Schrodinger-Maxwell Equations, Topol. Methods Nonlinear Anal., (1998), vol 11, 283-293.
P. dAvenia and G. Siciliano; Nonlinear Schrodinger equation in the Bopp-Podolsky electrodinamics: Solutions in the electrostatic case, J. Differential Equations, (2019), vol 269, 1025-1065.
G. M. Figueiredo, G. Siciliano; Multiple solutions for a Schrodinger-Bopp-Podolsky system with positive potentials, Math. Nachr., 296 (2023), 2332-2351 18 L. SORIANO H., G. SICILIANO EJDE-2023/??
D. Gilbarg, N. S. Trudinger; Elliptic Partial Differential Equations of Second Order, Springer, 1988.
E. Hebey; Electromagnetostatic study of the nonlinear Schrodinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting, Discrete Contin. Dyn. Syst., 39 (2019), 66836712.
L.-X. Huang, X.-P. Wu, C.-L. Tang; Multiple positive solutions for nonhomogeneous Schrodinger-Poisson systems with Berestycki-Lions type conditions, Electron. J. Differential Equations, 2021 (2021), no. 01, 1-14,
L. Li, P. Pucci, X.Tang; Ground state solutions for the nonlinear Schrodinger-Bopp-Podolski system with critical Sobolev exponent, Adv. Nonlinear Stud., 20 (2020), 511538.
B. Mascaro, G. Siciliano; Positive Solutions For a Schrodinger-Bopp-Podolsky system, Commun. Math., 31, no. 1 (2023), 237-249.
G. Ramos de Paula, G. Siciliano; Existence and limit behavior of least energy solutions to constrained Schrodinger-Bopp-Podolsky systems in R3, Z. Angew. Math. Phys., (2023) 74:56.
H. M. Santos Damian, G. Siciliano; Schrodinger-Bopp-Podolsky systems with vanishing potentials: small solutions and asymptotic behavior, preprint.
G. Siciliano, K. Silva; The fibering method approach for a non-linear Schrodinger equation coupled with the electromagnetic field, Publ. Mat., 64 (2020), 373390.
M. Struwe; Variational Methods and Their Applications to Non-linear Partial Differential Equations and Hamiltonian Systems, Springer, 1990.
Downloads
Published
License
Copyright (c) 2023 Lorena Soriano Hernandez, Gaetano Siciliano
This work is licensed under a Creative Commons Attribution 4.0 International License.
