No Access

Normalized solutions to a Schrödinger–Bopp–Podolsky system under Neumann boundary conditions

Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy

E-mail Address: gregorinafonso@mat.uniroma1.it

and

Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil

E-mail Address: sicilian@ime.usp.br

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219199721501005Cited by:10 (Source: Crossref)

Abstract

In this paper, we study a Schrödinger–Bopp–Podolsky (SBP) system of partial differential equations in a bounded and smooth domain of $ℝ^{3}$ with a nonconstant coupling factor. Under a compatibility condition on the boundary data we deduce existence of solutions by means of the Ljusternik–Schnirelmann theory.

Keywords:

AMSC: 35J50, 35J58, 35Q55, 35Q61

References

1. D. G. Afonso, Normalized solutions for a Schrödinger–Bopp–Podolsky system, M.Sc. dissertation, Universidade de São Paulo (2020). Google Scholar
2. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959) 623–727. Crossref, Web of Science, Google Scholar
3. V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998) 283–293. Crossref, Google Scholar
4. S. Chen and X. Tang, On the critical Schrödinger–Bopp–Podolsky system with general nonlinearities, Nonlinear Anal. 195 (2020) 111734. Crossref, Web of Science, Google Scholar
5. P. d’Avenia and G. Siciliano, Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: Solutions in the electrostatic case, J. Differential Equations 267 (2019) 1025–1065. Crossref, Web of Science, Google Scholar
6. G. M. Figueiredo and G. Siciliano, Multiple solutions for a Schrödinger–Bopp–Podolsky system with positive potentials, preprint (2020); arXiv:2006.12637. Google Scholar
7. F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems (Springer, 2010). Crossref, Google Scholar
8. E. Hebey, Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp–Podolsky electrodynamics in the Proca setting, Discrete Contin. Dyn. Syst. 39(11) (2019) 6683–6712. Crossref, Web of Science, Google Scholar
9. L. Li, P. Pucci and X. Tang, Ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent, Adv. Nonlinear Stud. 20(3) (2020) 511–538. Crossref, Web of Science, Google Scholar
10. B. Mascaro and G. Siciliano, Positive solutions for a Schrödinger–Bopp–Podolsky system in $ℝ^{3}$ , preprint (2020); arXiv:2009.08531. Google Scholar
11. L. Pisani and G. Siciliano, Neumann condition in the Schrödinger–Maxwell system, Topol. Methods Nonlinear Anal. 29 (2007) 251–264. Web of Science, Google Scholar
12. L. Pisani and G. Siciliano, Constrained Schrödinger–Poisson system with non-constant interaction, Commun. Contemp. Math. 15(1) (2013) 1250052. Link, Web of Science, Google Scholar
13. G. Siciliano and K. Silva, The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field, Publ. Mat. 64 (2020) 373–390. Crossref, Web of Science, Google Scholar
14. A. Szulkin, Ljusternik–Schnirelman theory on $C^{1}$ -manifolds, Ann. Inst. Henri Poincaré Anal. Non Linéaire 5(2) (1988) 119–139. Crossref, Web of Science, Google Scholar
15. M. E. Taylor, Partial Differential Equations I (Springer, 1996). Crossref, Google Scholar