On the planar Schrödinger-Poisson system with the axially symmetric potential. (English) Zbl 1431.35030
Summary: In this paper, we develop some new variational and analytic techniques to prove that the following planar Schrödinger-Poisson system
\[\begin{cases} -\Delta u + V(x) u + \phi u = f(u), & x \in \mathbb{R}^2, \\ \Delta \phi = u^2, & x \in \mathbb{R}^2, \end{cases}\]
admits a nontrivial solution and a ground state solution possessing the least energy in the axially symmetric functions space, where \(V(x)\) is axially symmetric. Our results improve and extend the ones in the case \(V = 1\) and \(f(u) = |u|^{p-2} u\) with \(2 < p < 6\). In particular, we use the assumption that \(2V(x) + \nabla V(x) \cdot x\) is bounded from below instead of the usually one that \(\lim_{|x| \to \infty} V(x) = 1\). Moreover, \(V(x)\) is even admitted to be unbounded.
\[\begin{cases} -\Delta u + V(x) u + \phi u = f(u), & x \in \mathbb{R}^2, \\ \Delta \phi = u^2, & x \in \mathbb{R}^2, \end{cases}\]
admits a nontrivial solution and a ground state solution possessing the least energy in the axially symmetric functions space, where \(V(x)\) is axially symmetric. Our results improve and extend the ones in the case \(V = 1\) and \(f(u) = |u|^{p-2} u\) with \(2 < p < 6\). In particular, we use the assumption that \(2V(x) + \nabla V(x) \cdot x\) is bounded from below instead of the usually one that \(\lim_{|x| \to \infty} V(x) = 1\). Moreover, \(V(x)\) is even admitted to be unbounded.
MSC:
| 35J47 | Second-order elliptic systems |
| 35J61 | Semilinear elliptic equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
References:
| [1] | Ambrosetti, A.; Ruiz, D., Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10, 391-404 (2008) · Zbl 1188.35171 |
| [2] | Azzollini, A., Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differ. Equ., 249, 1746-1763 (2010) · Zbl 1197.35096 |
| [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] | Benci, V.; Fortunato, D., Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14, 409-420 (2002) · Zbl 1037.35075 |
| [6] | Benguria, R.; Brezis, H.; Lieb, E., The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Commun. Math. Phys., 79, 167-180 (1981) · Zbl 0478.49035 |
| [7] | Catto, I.; Lions, P., Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Commun. Partial Differ. Equ., 18, 1149-1159 (1993) · Zbl 0807.35116 |
| [8] | Cerami, G.; Vaira, J., Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248, 521-543 (2010) · Zbl 1183.35109 |
| [9] | Chen, S. T.; Shi, J. P.; Tang, X. H., Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity, Discrete Contin. Dyn. Syst., Ser. A, 39, 5867-5889 (2019) · Zbl 1425.35014 |
| [10] | Chen, S. T.; Tang, X. H., Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system, Discrete Contin. Dyn. Syst., Ser. B, 24, 4685-4702 (2019) · Zbl 1429.35062 |
| [11] | Chen, S. T.; Tang, X. H., Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9, 496-515 (2020) · Zbl 1422.35023 |
| [12] | Cingolani, S.; Weth, T., On the planar Schrödinger-Poisson system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 169-197 (2016) · Zbl 1331.35126 |
| [13] | D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. R. Soc. Edinb. A, 134, 893-906 (2004) · Zbl 1064.35182 |
| [14] | D’Aprile, T.; Wei, J. C., Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differ. Equ., 25, 105 (2006) · Zbl 1207.35129 |
| [15] | D’Aprile, T.; Wei, J., Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 5, 219-259 (2006) · Zbl 1150.35006 |
| [16] | D’Aprile, T.; Wei, J., On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37, 321-342 (2005) · Zbl 1096.35017 |
| [17] | Du, M.; Weth, T., Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30, 3492-3515 (2017) · Zbl 1384.35010 |
| [18] | He, X. M., Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62, 869-889 (2011) · Zbl 1258.35170 |
| [19] | He, X. M.; Zou, W. M., Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53, Article 023702 pp. (2012) · Zbl 1274.81078 |
| [20] | Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 1633-1659 (1997) · Zbl 0877.35091 |
| [21] | Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \(R^N\), Proc. R. Soc. Edinb. A, 129, 787-809 (1999) · Zbl 0935.35044 |
| [22] | Li, G. B.; Peng, S. J.; Yan, S. S., Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12, 1069-1092 (2010) · Zbl 1206.35082 |
| [23] | Lieb, E., Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. Math., 118, 349-374 (1983) · Zbl 0527.42011 |
| [24] | Lieb, E.; Loss, M., Analysis, Graduate Studies in Mathematics, vol. 14 (2001), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0966.26002 |
| [25] | Lieb, E. H., Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53, 263-301 (1981) · Zbl 1114.81336 |
| [26] | Lions, P., Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109, 33-97 (1984) · Zbl 0618.35111 |
| [27] | Liu, Z. S.; Zhang, Z. T.; Huang, S. B., Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differ. Equ., 266, 5912-5941 (2019) · Zbl 1421.35156 |
| [28] | Markowich, P. A.; Ringhofer, C. A.; Schmeiser, C., Semiconductor Equations (1990), x+248 · Zbl 0765.35001 |
| [29] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics, vol. III: Scattering Theory (1979), Academic Press: Academic Press New York · Zbl 0405.47007 |
| [30] | Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674 (2006) · Zbl 1136.35037 |
| [31] | Stubbe, J., Bound states of two-dimensional Schrödinger-Newton equations, eprint |
| [32] | Sun, J. J.; Ma, S. M., Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differ. Equ., 260, 2119-2149 (2016) · Zbl 1334.35044 |
| [33] | Tang, X. H., Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58, 715-728 (2015) · Zbl 1321.35055 |
| [34] | Tang, X. H.; Chen, S. T., Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differ. Equ., 56, 110 (2017) · Zbl 1376.35056 |
| [35] | Tang, X. H.; Chen, S. T., Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., Ser. A, 37, 4973-5002 (2017) · Zbl 1371.35051 |
| [36] | Tang, X. H.; Chen, S. T., Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal., 9, 413-437 (2020) · Zbl 1421.35068 |
| [37] | Willem, M., Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24 (1996), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0856.49001 |
| [38] | Zhao, L.; Zhao, F., On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346, 155-169 (2008) · Zbl 1159.35017 |
| [39] | Zhang, J. J., The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth, J. Math. Phys., 55, Article 031507 pp. (2014) · Zbl 1286.81075 |
| [40] | Zhang, J. J.; Xu, Z. J.; Zou, W. M., Standing waves for nonlinear Schrödinger equations involving critical growth, J. Lond. Math. Soc., 90, 827-844 (2014) · Zbl 1317.35247 |
| [41] | Zhao, L. G.; Zhao, F. K., Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70, 2150-2164 (2009) · Zbl 1156.35374 |
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