Multiple positive solutions for a Schrödinger–Poisson system with critical and supercritical growths

Обложка

Цитировать

Полный текст

Открытый доступ Открытый доступ
Доступ закрыт Доступ предоставлен
Доступ закрыт Только для подписчиков

Аннотация

In this paper, we are concerned with the following Schrödinger–Poisson system$$\begin{cases}-\Delta u+u+\lambda\phi u= Q(x)|u|^{4}u+\mu\dfrac{|x|^\beta}{1+|x|^\beta}|u|^{q-2}u&in \mathbb{R}^3,-\Delta \phi=u^{2} &in \mathbb{R}^3,\end{cases}$$where $0< \beta<3$, $60$ are real parameters. By the variational method and the Nehari method, we obtain that the system has $k$ positive solutions.Bibliography: 31 titles.

Об авторах

Jun Lei

Guizhou University

Email: gzmysxx88@sina.com

PhD, без звания

Hongmin Suo

Guizhou Minzu University

Автор, ответственный за переписку.
Email: gzmysxx88@sina.com

Список литературы

  1. V. Benci, D. Fortunato, “An eigenvalue problem for the Schrödinger–Maxwell equations”, Topol. Methods Nonlinear Anal., 11:2 (1998), 283–293
  2. O. Bokanowski, J. L. Lopez, J. Soler, “On an exchange interaction model for quantum transport: the Schrödinger–Poisson–Slater system”, Math. Models Methods Appl. Sci., 13:10 (2003), 1397–1412
  3. D. Ruiz, “The Schrödinger–Poisson equation under the effect of a nonlinear local term”, J. Funct. Anal., 237:2 (2006), 655–674
  4. I. V. Barashenkov, A. D. Gocheva, V. G. Makhankov, I. V. Puzynin, “Stability of the soliton-like “bubbles””, Phys. D, 34:1-2 (1989), 240–254
  5. В. Г. Картавенко, “Решения солитонного типа в ядерной гидродинамике”, Ядерная физика, 40 (1984), 377–388
  6. V. Benci, D. Fortunato, “Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations”, Rev. Math. Phys., 14:4 (2002), 409–420
  7. Ching-yu Chen, Yueh-cheng Kuo, Tsung-fang Wu, “Existence and multiplicity of positive solutions for the nonlinear Schrödinger–Poisson equations”, Proc. Roy. Soc. Edinburgh Sect. A, 143:4 (2013), 745–764
  8. Fuyi Li, Yuhua Li, Junping Shi, “Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent”, Commun. Contemp. Math., 16:6 (2014), 1450036, 28 pp.
  9. Gongbao Li, Shuangjie Peng, Shusen Yan, “Infinitely many positive solutions for the nonlinear Schrödinger–Poisson system”, Commun. Contemp. Math., 12:6 (2010), 1069–1092
  10. Haidong Liu, “Positive solutions of an asymptotically periodic Schrödinger–Poisson system with critical exponent”, Nonlinear Anal. Real World Appl., 32 (2016), 198–212
  11. Juntao Sun, Tsung-fang Wu, Zhaosheng Feng, “Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system”, J. Differential equations, 260:1 (2016), 586–627
  12. A. Azzollini, “Concentration and compactness in nonlinear Schrödinger–Poisson system with a general nonlinearity”, J. Differential Equations, 249:7 (2010), 1746–1763
  13. A. Ambrosetti, D. Ruiz, “Multiple bound states for the Schrödinger–Poisson problem”, Commun. Contemp. Math., 10:3 (2008), 391–404
  14. Yuhua Li, Hua Gu, “Existence of solutions to Schrödinger–Poisson systems with critical and supercritical nonlinear terms”, Math. Methods Appl. Sci., 42:7 (2019), 2279–2286
  15. A. Ambrosetti, “On Schrödinger–Poisson systems”, Milan. J. Math., 76 (2008), 257–274
  16. A. Azzollini, P. d'Avenia, “On a system involving a critically growing nonlinearity”, J. Math. Anal. Appl., 387:1 (2012), 433–438
  17. C. O. Alves, M. A. S. Souto, S. H. M. Soares, “Schrödinger–Poisson equations without Ambrosetti–Rabinowitz condition”, J. Math. Anal. Appl., 377:2 (2011), 584–592
  18. Zhisu Liu, Shangjiang Guo, “On ground state solutions for the Schrödinger–Poisson equations with critical growth”, J. Math. Anal. Appl., 412:1 (2014), 435–448
  19. Jian Zhang, “On the Schrödinger–Poisson equations with a general nonlinearity in the critical growth”, Nonlinear Anal., 75:18 (2012), 6391–6401
  20. Jian Zhang, “On ground state and nodal solutions of Schrödinger–Poisson equations with critical growth”, J. Math. Anal. Appl., 428:1 (2015), 387–404
  21. Lirong Huang, E. M. Rocha, Jianqing Chen, “Positive and sign-changing solutions of a Schrödinger–Poisson system involving a critical nonlinearity”, J. Math. Anal. Appl., 408:1 (2013), 55–69
  22. Leiga Zhao, Fukun Zhao, “Positive solutions for Schrödinger–Poisson equations with a critical exponent”, Nonlinear Anal., 70:6 (2009), 2150–2164
  23. Daomin Cao, J. Chabrowski, “Multiple solutions of nonhomogeneous elliptic equation with critical nonlinearity”, Differential Integral Equations, 10:5 (1997), 797–814
  24. Daomin Cao, E. S. Noussair, “Multiple positive and nodal solutions for semilinear elliptic problems with critical exponents”, Indiana Univ. Math. J., 44:4 (1995), 1249–1271
  25. Pigong Han, “Multiple solutions to singular critical elliptic equations”, Israel J. Math., 156 (2006), 359–380
  26. Yi Sheng Huang, “Multiple positive solutions of nonhomogeneous equations involving the $p$-Laplacian”, Nonlinear Anal., 43:7 (2001), 905–922
  27. Huei-li Lin, “Positive solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent”, Nonlinear Anal., 75:4 (2012), 2660–2671
  28. Jia-feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang, “Existence and multiplicity of positive solutions for a class of elliptic equations involving critical Sobolev exponents”, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 110:2 (2016), 483–501
  29. H. Brezis, L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents”, Comm. Pure. Appl. Math., 36:4 (1983), 437–477
  30. P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The limit case. I”, Rev. Mat. Iberoamericana, 1:1 (1985), 145–201
  31. I. Ekeland, “On the variational principle”, J. Math. Anal. Appl., 48:2 (1974), 324–353

© Рэ Ч., Suo H., 2023

Данный сайт использует cookie-файлы

Продолжая использовать наш сайт, вы даете согласие на обработку файлов cookie, которые обеспечивают правильную работу сайта.

О куки-файлах