In this paper, we study the following kind of Schrödinger-Poisson system in $ { \mathbb{R}}^{2} $
$ \begin{equation*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi u = K(x)f(u), \ \ \ x\in{ \mathbb{R}}^{2}, \\ \Delta \phi = u^{2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in{ \mathbb{R}}^{2}, \end{array}\right. \end{equation*} $
where $ f\in C({ \mathbb{R}}, { \mathbb{R}}) $, $ V(x) $ and $ K(x) $ are both axially symmetric functions. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution for the above planar system. Our result improves and extends the existing works.
Citation: Qiongfen Zhang, Kai Chen, Shuqin Liu, Jinmei Fan. Existence of axially symmetric solutions for a kind of planar Schrödinger-Poisson system[J]. AIMS Mathematics, 2021, 6(7): 7833-7844. doi: 10.3934/math.2021455
In this paper, we study the following kind of Schrödinger-Poisson system in $ { \mathbb{R}}^{2} $
$ \begin{equation*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi u = K(x)f(u), \ \ \ x\in{ \mathbb{R}}^{2}, \\ \Delta \phi = u^{2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in{ \mathbb{R}}^{2}, \end{array}\right. \end{equation*} $
where $ f\in C({ \mathbb{R}}, { \mathbb{R}}) $, $ V(x) $ and $ K(x) $ are both axially symmetric functions. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution for the above planar system. Our result improves and extends the existing works.
| [1] |
S. Chen, X. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differ. Equations, 268 (2020), 945–976. doi: 10.1016/j.jde.2019.08.036
|
| [2] |
R. Benguria, H. Brezis, E. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167–180. doi: 10.1007/BF01942059
|
| [3] |
A. Paredes, D. Olivieri, M. Humberto, From optics to dark matter: A review on nonlinear Schrödinger-Poisson systems, Physica D, 403 (2020), 132301. doi: 10.1016/j.physd.2019.132301
|
| [4] | J. Chen, S. Chen, X. Tang, Ground state solutions for asymptotically periodic Schrödinger-Poisson systems in ${ \mathbb{R}}^{2}$, Electron. J. Differ. Equations, 192 (2018), 1–18. |
| [5] | X. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715–728. |
| [6] | F. Bernini, D. Mugnai, On a logarithmic Hartree equation, Adv. Nonlinear Anal., 9 (2020), 850–865. |
| [7] |
S. Cingolani, T. Weth, On the planar Schrödinger-Poisson system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33 (2016), 169–197. doi: 10.1016/j.anihpc.2014.09.008
|
| [8] |
M. Du, T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492–3515. doi: 10.1088/1361-6544/aa7eac
|
| [9] |
S. Chen, J. Shi, X. Tang, Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity, Discrete Contin. Dyn. Syst., Ser. A, 39 (2019), 5867–5889. doi: 10.3934/dcds.2019257
|
| [10] | S. Chen, X. Tang, Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system, Discrete Contin. Dyn. Syst., Ser. B, 24 (2019), 4685–4702. |
| [11] |
L. Wen, S. Chen, V. D. Rădulescu, Axially symmetric solutions of Schrödinger-Poisson system with zero mass potential in ${ \mathbb{R}}^{2}$, Appl. Math. Lett., 104 (2020), 106244. doi: 10.1016/j.aml.2020.106244
|
| [12] |
Z. Chen, W. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Func. Anal., 262 (2012), 3091–3107. doi: 10.1016/j.jfa.2012.01.001
|
| [13] | S. Chen, X. Tang, Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9 (2020), 496–515. |
| [14] |
J. Peng, S. Chen, X. Tang, Semiclassical solutions for linearly coupled Schrödinger equations without compactness, Complex Var. Elliptic Equations, 64 (2019), 548–556. doi: 10.1080/17476933.2018.1450395
|
| [15] |
X. Tang, New super-quadratic conditions for asymptotically periodic Schrödinger equations, Can. Math. Bull., 60 (2017), 422–435. doi: 10.4153/CMB-2016-090-2
|
| [16] |
X. Tang, X. Lin, J. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equations, 31 (2019), 369–383. doi: 10.1007/s10884-018-9662-2
|
| [17] | G. Che, H. Chen, Existence of multiple nontrivial solutions for a class of quasilinear Schrödinger equations on ${ \mathbb{R}}^{N}$, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 39–53. |
| [18] |
X. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104–116. doi: 10.1017/S144678871400041X
|
| [19] | B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in ${ \mathbb{R}}^{N}$, Ann. Mat. Pura Appl., 183 (2002), 73–83. |
| [20] | J. Fan, Y. Jiang, Q. Zhang, Semiclassical Solutions for a Kind of Coupled Schrödinger Equations, Adv. Math. Phys., 2020 (2020), 4378691. |
| [21] |
X. Zhang, Existence and multiplicity of solutions for a class of elliptic boundary value problems, J. Math. Anal. Appl., 410 (2014), 213–226. doi: 10.1016/j.jmaa.2013.08.001
|
| [22] |
Q. Zhang, C. Gan, T. Xiao, Z. Jia, Some results of nontrivial solutions for Klein-Gordon-Maxwell systems with local super-quadratic conditions, J. Geom. Anal., 31 (2021), 5372–5394. doi: 10.1007/s12220-020-00483-2
|
| [23] | L. Wang, X. Zhang, H. Fang, Multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, Taiwan. J. Math., 21 (2017), 881–912. |
| [24] | Q. Zhang, C. Gan, T. Xiao, Z. Jia, An improved result for Klein-Gordon-Maxwell systems with steep potential well, Math. Methods Appl. Sci., 2020 (2020). |
| [25] | E. Lieb, M. Loss, Analysis, 2nd Ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. |
Qiongfen Zhang, Kai Chen, Shuqin Liu, Jinmei Fan. Existence of axially symmetric solutions for a kind of planar Schrödinger-Poisson system[J]. AIMS Mathematics, 2021, 6(7): 7833-7844. doi: 10.3934/math.2021455
DownLoad: