On quasilinear Schrödinger-Poisson system involving Berestycki-Lions type conditions. (English) Zbl 07829766
Summary: In this work we study a quasilinear Schrödinger-Poisson system which is coupled by a Schrödinger equation of \(p\)-Laplacian and a Poisson equation of \(q\)-Laplacian, with a general nonlinear term. The nonlinearity satisfies the Berestycki-Lions type conditions. By means of variational methods, we get the existence of nontrivial solutions for the quasilinear system.
MSC:
| 35J47 | Second-order elliptic systems |
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
References:
| [1] | Azzollini, A., Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differ. Equ., 249, 1746-1763, 2010 · Zbl 1197.35096 |
| [2] | Azzollini, A.; d’Avenia, P.; Pomponio, A., On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 27, 779-791, 2010 · Zbl 1187.35231 |
| [3] | Azzollini, A.; Pomponio, A., Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345, 90-108, 2008 · Zbl 1147.35091 |
| [4] | Benci, V.; Fortunato, D., An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11, 283-293, 1998 · Zbl 0926.35125 |
| [5] | Benedikt, J.; Girg, P.; Kotrla, L.; Takáč, P., Origin of the p-Laplacian and A. Missbach, Electron. J. Differ. Equ., 16, 2018 · Zbl 1386.76158 |
| [6] | Benguria, R.; Brézis, H.; Lieb, E. H., The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Commun. Math. Phys., 79, 167-180, 1981 · Zbl 0478.49035 |
| [7] | Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345, 1983 · Zbl 0533.35029 |
| [8] | Catto, I.; Lions, P. L., Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I. A necessary and sufficient condition for the stability of general molecular system, Commun. Partial Differ. Equ., 17, 1051-1110, 1992 · Zbl 0767.35065 |
| [9] | D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. R. Soc. Edinb., Sect. A, 134, 893-906, 2004 · Zbl 1064.35182 |
| [10] | D’Aprile, T.; Mugnai, D., Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4, 307-322, 2004 · Zbl 1142.35406 |
| [11] | d’Avenia, P., Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2, 177-192, 2002 · Zbl 1007.35090 |
| [12] | Du, Y.; Su, J.; Wang, C., The Schrödinger-Poisson system with p-Laplacian, Appl. Math. Lett., 120, Article 107286 pp., 2021 · Zbl 1479.35335 |
| [13] | Du, Y.; Su, J.; Wang, C., On a quasilinear Schrödinger-Poisson system, J. Math. Anal. Appl., 505, Article 125446 pp., 2022 · Zbl 1479.35336 |
| [14] | Du, Y.; Su, J.; Wang, C., The quasilinear Schrödinger-Poisson system, J. Math. Phys., 64, Article 071502 pp., 2023 · Zbl 1520.35141 |
| [15] | Huang, L.-X.; Wu, X.-P.; Tang, C.-L., Multiple positive solutions for nonhomogeneous Schrödinger-Poisson systems with Berestycki-Lions type conditions, Electron. J. Differ. Equ., Article 1 pp., 2021 · Zbl 1458.35143 |
| [16] | Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 1633-1659, 1997 · Zbl 0877.35091 |
| [17] | Jeanjean, L.; Le Coz, S., An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differ. Equ., 11, 813-840, 2006 · Zbl 1155.35095 |
| [18] | Kikuchi, H., On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67, 1445-1456, 2007 · Zbl 1119.35085 |
| [19] | Lieb, E. H., Thomas-Fermi and related theories and molecules, Rev. Mod. Phys., 53, 603-641, 1981 · Zbl 1114.81336 |
| [20] | Lions, P. L., Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109, 33-97, 1987 · Zbl 0618.35111 |
| [21] | Markowich, P.; Ringhofer, C.; Schmeiser, C., Semiconductor Equations, 1990, Springer-Verlag: Springer-Verlag New York · Zbl 0765.35001 |
| [22] | Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674, 2006 · Zbl 1136.35037 |
| [23] | Sun, J. J.; Ma, S. W., Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differ. Equ., 260, 2119-2149, 2016 · Zbl 1334.35044 |
| [24] | Willem, M., Minimax Theorems, 1996, Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston · Zbl 0856.49001 |
| [25] | Willem, M., Functional Analysis. Fundamentals and Applications, 2013, Birkhäuser/Springer: Birkhäuser/Springer New York · Zbl 1284.46001 |
| [26] | Yin, L.-F.; Wu, X.-P.; Tang, C.-L., Ground state solutions for an asymptotically 2-linear Schrödinger-Poisson system, Appl. Math. Lett., 87, 7-12, 2019 · Zbl 1411.35089 |
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.
