Academic Journal of Mathematical Sciences, 2025, 6(2); doi: 10.25236/AJMS.2025.060215.
Shuwen He1,2
1School of Mathematics, Physics and Statistics, Sichuan Minzu College, Kangding, 626001, China
2Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu, 610068, China
This article is to study a Schrödinger-Bopp-Podolsky system has steep potential well and concave-convex nonlinearities. This system is a coupled system that describes physical phenomena such as charge motion in physics, and its mathematical research involves the existence and asymptotic behavior of solutions to nonlinear partial differential equations. By combining the variational methods and the truncation technique to prove the existence and concentration behavior results of nontrivial solutions for this system. Specifically, first the defined truncation function is merged into the convolution term of the corresponding energy functional of the system and it is analyzed that the Gerami sequence of the energy functional is norm bounded. Next, this sequence is proved to be a norm bounded by the energy functional of the original system. Finally, it is proved that there are strong convergent subsequences in the obtained bounded sequence, and the main results are displayed using standard analysis methods. These results of the article have been extensively improved and expanded to previous works.
Schrödinger-Bopp-Podolsky System, Variational Methods, Truncation Technique
Shuwen He. Results on Nontrivial Solutions of the Schrödinger-Bopp-Podolsky System. Academic Journal of Mathematical Sciences (2025), Vol. 6, Issue 2: 113-120. https://doi.org/10.25236/AJMS.2025.060215.
[1] Bopp F. Eine lineare theorie des elektrons[J]. Annalen der Physik, 1940, 430(5): 345-384.
[2] Podolsky B. A generalized electrodynamics part I-non-quantum[J]. Physical Review, 1942, 62 (1-2): 68-71.
[3] Yang J, Chen H, Liu S. The existence of nontrivial solution of a class of Schrödinger-Bopp-Podolsky system with critical growth[J]. Boundary Value Problems, 2020, 2020(1): 144.
[4] Wang L, Chen H, Liu S. Existence and multiplicity of sign-changing solutions for a Schrödinger-Bopp-Podolsky system[J], Topological Methods in Nonlinear Analysis, 2022, (59): 913-940.
[5] Zhang Q. Sign-changing solutions for Schrödinger-Bopp-Podolsky system with general nonlinearity[J]. Zeitschrift für angewandte Mathematik und Physik, 2022, 73(6): 235.
[6] Peng X. Existence and multiplicity of solutions for the Schrödinger-Bopp-Podolsky system[J]. Bulletin of the Malaysian Mathematical Sciences Society, 2022, 45(6): 3423-3468.
[7] Zhu Q, Chen C, Yuan C. Schrödinger-Bopp-Podolsky System with Steep Potential Well[J]. Qualitative Theory of Dynamical Systems, 2023, 22(4): 140.
[8] Ekeland I. Convexity methods in Hamiltonian mechanics[M]. Springer Science & Business Media, 2012.
[9] Willem M. Minimax Theorems[M]. Birkhäuser, Boston, 1996.
[10] He S, Wen X. Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well[J]. AIMS Mathematics, 2023, 8(3): 6432-6446.