The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth

Wenbo Wang; Wei Zhang; Yongkun Li

doi:doi:10.11648/j.pamj.20200906.13

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The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth

Wenbo Wang, Wei Zhang, Yongkun Li

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 6)

Received: 8 February 2020 Accepted: 8 September 2020 Published: 4 December 2020

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Abstract

In 1990, the notion of critical growth in ℝ² was introduced by Adimurthi and Yadava. Soon afterwards, de Figueiredo, Miyagaki and Ruf also studied the solvability of elliptic equations in dimension two. They treated the problems in the subcritical and the critical case. In 2019, Alves and Figueieredo proved the existence of a positive solution for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponent critical growth. Also, in 2020, Chen and Tang investigated the planar Schrodinger-Poisson system with the critical growth nonlinearity. Under the axially symmetric assumptions, they obtained infinitely many pairs solutions and ground states. In this work, motivated by the works mentioned above and Z. Angew. Math. Phys., 66 (2015) 3267-3282, Z. Angew. Math. Phys., 67 (2016) 102, 18, we study the planar Schrödinger-Newton system with a Coulomb potential where the nonlinearity f is autonomous nonlinearity which belongs to C¹ and satisfies super-linear at zero and exponential critical at infinity. Moreover, we need that f satisfies the Nehari type monotonic condition. We obtain a least-energy sign-changing solution via the variational method. To be more precise, we define the sign-changing Nehari manifold. And the least-energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 6)
DOI	10.11648/j.pamj.20200906.13
Page(s)	118-123
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Schrödinger-Newton System, The Exponential Critical Growth, Sign-Changing Solutions, Sign-Changing Nehari Manifold

References

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[2]	C. O. Alves, G. M. Figueieredo, Existence of positive solution for a planar Schrödinger-Poisson system with exponential growth, J. Math. Phys., 60 (2019) 011503.
[3]	V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998) 283-293.
[4]	D. Cao, Nontrivial solutions of semilinear elliptic equation with critical exponent in ℝ², Comm. Partial Differential Equations, 17 (1992) 407-435.
[5]	S. Chen, X. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson system type problems in ℝ³, Z. Angew. Math. Phys., 67 (2016) 102, 18 pp.
[6]	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., 39 (2019) 5867-5889.
[7]	S. Chen, X. Tang, Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system, Discrete Contin. Dyn. Syst. Ser. B, 24(9)(2019) 4685-4702.
[8]	S. Chen, X. Tang, On the planar Schrödinger- Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020) 945-976.
[9]	S. Chen, X. Tang, Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth, J. Differential Equations, 269 (2020) 9144-9174.
[10]	P. Choquard, J. Stubbe, M. Vuffray,Stationary solutions of the Schrödinger Newton model an ODE approach, Differ. Integral Equ., 21 (2008) 665-679.
[11]	S. Cingolani and L. Jeanjean, Stationary waves with prescribed L²-norm for the planar Schrödinger-Poisson system, SIAM J. Math. Anal., 51 (2019) 3533-3568.
[12]	S. Cingolani, T. Weth, On the Schrödinger-Poisson system, Ann. Inst. H. Poincar_e Anal. Non Linéaire, 33 (2016) 169-197.
[13]	D.G. de Figueiredo, O.H. Miyagaki, B. Ruf, Elliptic equations in ℝ² with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995) 139-153.
[14]	J.M. Bezerra Doó, N-Laplacian equations in ℝ^N with critical growth, Abstr. Appl. Anal., 2 (1997) 301-315.
[15]	M. Du, T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017) 3492-3515.
[16]	X. He, W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012) 143-162.
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[18]	W. Shuai, Q. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger- Poisson system in ℝ³, Z. Angew. Math. Phys., 66 (2015) 3267-3282.
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APA Style

Wenbo Wang, Wei Zhang, Yongkun Li. (2020). The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth. Pure and Applied Mathematics Journal, 9(6), 118-123. https://doi.org/10.11648/j.pamj.20200906.13

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ACS Style

Wenbo Wang; Wei Zhang; Yongkun Li. The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth. Pure Appl. Math. J. 2020, 9(6), 118-123. doi: 10.11648/j.pamj.20200906.13

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AMA Style

Wenbo Wang, Wei Zhang, Yongkun Li. The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth. Pure Appl Math J. 2020;9(6):118-123. doi: 10.11648/j.pamj.20200906.13

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@article{10.11648/j.pamj.20200906.13,
  author = {Wenbo Wang and Wei Zhang and Yongkun Li},
  title = {The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {6},
  pages = {118-123},
  doi = {10.11648/j.pamj.20200906.13},
  url = {https://doi.org/10.11648/j.pamj.20200906.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200906.13},
  abstract = {In 1990, the notion of critical growth in ℝ2 was introduced by Adimurthi and Yadava. Soon afterwards, de Figueiredo, Miyagaki and Ruf also studied the solvability of elliptic equations in dimension two. They treated the problems in the subcritical and the critical case. In 2019, Alves and Figueieredo proved the existence of a positive solution for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponent critical growth. Also, in 2020, Chen and Tang investigated the planar Schrodinger-Poisson system with the critical growth nonlinearity. Under the axially symmetric assumptions, they obtained infinitely many pairs solutions and ground states. In this work, motivated by the works mentioned above and Z. Angew. Math. Phys., 66 (2015) 3267-3282, Z. Angew. Math. Phys., 67 (2016) 102, 18, we study the planar Schrödinger-Newton system with a Coulomb potential where the nonlinearity f is autonomous nonlinearity which belongs to C1 and satisfies super-linear at zero and exponential critical at infinity. Moreover, we need that f satisfies the Nehari type monotonic condition. We obtain a least-energy sign-changing solution via the variational method. To be more precise, we define the sign-changing Nehari manifold. And the least-energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.},
 year = {2020}
}

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TY - JOUR
T1 - The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth
AU - Wenbo Wang
AU - Wei Zhang
AU - Yongkun Li
Y1 - 2020/12/04
PY - 2020
N1 - https://doi.org/10.11648/j.pamj.20200906.13
DO - 10.11648/j.pamj.20200906.13
T2 - Pure and Applied Mathematics Journal
JF - Pure and Applied Mathematics Journal
JO - Pure and Applied Mathematics Journal
SP - 118
EP - 123
PB - Science Publishing Group
SN - 2326-9812
UR - https://doi.org/10.11648/j.pamj.20200906.13
AB - In 1990, the notion of critical growth in ℝ2 was introduced by Adimurthi and Yadava. Soon afterwards, de Figueiredo, Miyagaki and Ruf also studied the solvability of elliptic equations in dimension two. They treated the problems in the subcritical and the critical case. In 2019, Alves and Figueieredo proved the existence of a positive solution for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponent critical growth. Also, in 2020, Chen and Tang investigated the planar Schrodinger-Poisson system with the critical growth nonlinearity. Under the axially symmetric assumptions, they obtained infinitely many pairs solutions and ground states. In this work, motivated by the works mentioned above and Z. Angew. Math. Phys., 66 (2015) 3267-3282, Z. Angew. Math. Phys., 67 (2016) 102, 18, we study the planar Schrödinger-Newton system with a Coulomb potential where the nonlinearity f is autonomous nonlinearity which belongs to C1 and satisfies super-linear at zero and exponential critical at infinity. Moreover, we need that f satisfies the Nehari type monotonic condition. We obtain a least-energy sign-changing solution via the variational method. To be more precise, we define the sign-changing Nehari manifold. And the least-energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.
VL - 9
IS - 6
ER -

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Author Information

Wenbo Wang

School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan, P.R. China
Wei Zhang

School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, P.R. China
Yongkun Li

School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan, P.R. China

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APA Style

Wenbo Wang, Wei Zhang, Yongkun Li. (2020). The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth. Pure and Applied Mathematics Journal, 9(6), 118-123. https://doi.org/10.11648/j.pamj.20200906.13

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ACS Style

Wenbo Wang; Wei Zhang; Yongkun Li. The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth. Pure Appl. Math. J. 2020, 9(6), 118-123. doi: 10.11648/j.pamj.20200906.13

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AMA Style

Wenbo Wang, Wei Zhang, Yongkun Li. The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth. Pure Appl Math J. 2020;9(6):118-123. doi: 10.11648/j.pamj.20200906.13

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@article{10.11648/j.pamj.20200906.13,
  author = {Wenbo Wang and Wei Zhang and Yongkun Li},
  title = {The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {6},
  pages = {118-123},
  doi = {10.11648/j.pamj.20200906.13},
  url = {https://doi.org/10.11648/j.pamj.20200906.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200906.13},
  abstract = {In 1990, the notion of critical growth in ℝ2 was introduced by Adimurthi and Yadava. Soon afterwards, de Figueiredo, Miyagaki and Ruf also studied the solvability of elliptic equations in dimension two. They treated the problems in the subcritical and the critical case. In 2019, Alves and Figueieredo proved the existence of a positive solution for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponent critical growth. Also, in 2020, Chen and Tang investigated the planar Schrodinger-Poisson system with the critical growth nonlinearity. Under the axially symmetric assumptions, they obtained infinitely many pairs solutions and ground states. In this work, motivated by the works mentioned above and Z. Angew. Math. Phys., 66 (2015) 3267-3282, Z. Angew. Math. Phys., 67 (2016) 102, 18, we study the planar Schrödinger-Newton system with a Coulomb potential where the nonlinearity f is autonomous nonlinearity which belongs to C1 and satisfies super-linear at zero and exponential critical at infinity. Moreover, we need that f satisfies the Nehari type monotonic condition. We obtain a least-energy sign-changing solution via the variational method. To be more precise, we define the sign-changing Nehari manifold. And the least-energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.},
 year = {2020}
}

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