The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field. (English) Zbl 1448.35171

The paper is concerned with the Schrödinger-Bopp-Podolsky system in \(\mathbb{R}^3\) \[\left\{\begin{array}{l} -\Delta u+\omega u+q^2 \phi u=|u|^{p-2}u,\\ -\Delta \phi+a^2\Delta^2\phi=4\pi u^2, \end{array}\right.\tag{1}\] where \(\omega>0\), \(a\ge 0\), \(p\in (2,3]\) and \(q>0\). By using the fibering method, the authors prove that system (1) has no nontrivial solutions for \(q\) large, and it has two radial solutions for \(q\) small. Some qualitative properties for the energy level of solutions and variational characterizations of the extremal values of \(q\) are also presented.


35J47 Second-order elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI arXiv Euclid


[1] V. Benci and D. Fortunato, An eigenvalue problem for the Schr¨odinger- Maxwell equations,Topol. Methods Nonlinear Anal.11(2)(1998), 283-293. DOI: 10.12775/TMNA.1998.019. · Zbl 0926.35125
[2] P. d’Avenia and G. Siciliano, Nonlinear Schr¨odinger equation in the Bopp- Podolsky electrodynamics:Solutions in the electrostatic case,J. Differential Equations267(2)(2019), 1025-1065.DOI: 10.1016/j.jde.2019.02.001. · Zbl 1432.35080
[3] Y. Ilyasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient,Topol. Methods Nonlinear Anal.49(2)(2017), 683-714. DOI: 10.12775/TMNA.2017.005. · Zbl 1375.35158
[4] Y. Ilyasov and K. Silva, On branches of positive solutions forp-Laplacian problems at the extreme value of the Nehari manifold method,Proc. Amer. Math. Soc.146(7)(2018), 2925-2935.DOI: 10.1090/proc/13972. · Zbl 1435.35185
[5] D. Ruiz, The Schr¨odinger-Poisson equation under the effect of a nonlinear local term,J. Funct. Anal.237(2)(2006), 655-674.DOI: 10.1016/j.jfa.2006.04.005. · Zbl 1136.35037
[6] D. Ruiz and G. Siciliano, A note on the Schr¨odinger-Poisson-Slater equation on bounded domains,Adv. Nonlinear Stud.8(1)(2008), 179-190.DOI: 10.1515/ ans-2008-0106. · Zbl 1160.35020
[7] K. Silva and A. Macedo, Local minimizers over the Nehari manifold for a class of concave-convex problems with sign changing nonlinearity,J. Differential Equations265(5)(2018), 1894-1921.DOI: 10.1016/j.jde.2018.04.018. · Zbl 1392.35172
[8] W.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.