대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제37권4호
  • /
  • Pages.1269-1284
  • /
  • 2022
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

GROUND STATE SIGN-CHANGING SOLUTIONS FOR NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH INDEFINITE POTENTIALS

  • Yu, Shubin (School of Mathematical Sciences TianGong University) ;
  • Zhang, Ziheng (School of Mathematical Sciences TianGong University)
  • 투고 : 2021.10.14
  • 심사 : 2022.01.11
  • 발행 : 2022.10.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

This paper is concerned with the following Schrödinger-Poisson system $$\{\begin{array}{lll}-{\Delta}u+V(x)u+K(x){\phi}u=a(x){\mid}u{\mid}^{p-2}u&&\text{ in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=K(x)u^2&&\text{ in }{\mathbb{R}}^3,\end{array}$$ where 4 < p < 6. For the case that K is nonnegative, V and a are indefinite, we prove the above problem possesses one ground state sign-changing solution with exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions is larger than that of the ground state solutions. The novelty of this paper is that the potential a is indefinite and allowed to vanish at infinity. In this sense, we complement the existing results obtained by Batista and Furtado [5].

키워드

과제정보

This work was financially supported by the National Natural Science Foundation of China (Grant No.11771044 and No.12171039).

참고문헌

  1. A. Ambrosetti, On Schrodinger-Poisson systems, Milan J. Math. 76 (2008), 257-274. https://doi.org/10.1007/s00032-008-0094-z
  2. A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrodinger-Poisson problem, Commun. Contemp. Math. 10 (2008), no. 3, 391-404. https://doi.org/10.1142/S021919970800282X
  3. A. Azzollini, P. d'Avenia, and A. Pomponio, On the Schrodinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010), no. 2, 779-791. https://doi.org/10.1016/j.anihpc.2009.11.012
  4. A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear SchrodingerMaxwell equations, J. Math. Anal. Appl. 345 (2008), no. 1, 90-108. https://doi.org/10.1016/j.jmaa.2008.03.057
  5. A. M. Batista and M. F. Furtado, Positive and nodal solutions for a nonlinear Schrodinger-Poisson system with sign-changing potentials, Nonlinear Anal. Real World Appl. 39 (2018), 142-156. https://doi.org/10.1016/j.nonrwa.2017.06.005
  6. G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrodinger-Poisson systems, J. Differential Equations 248 (2010), no. 3, 521-543. https://doi.org/10.1016/j.jde.2009.06.017
  7. G. Chai and W. Liu, Least energy sign-changing solutions for Kirchhoff-Poisson systems, Bound. Value Probl. 2019 (2019), Paper No. 160, 25 pp. https://doi.org/10.1186/s13661-019-1280-3
  8. C. Chen, Y. Kuo, and T. Wu, Existence and multiplicity of positive solutions for the nonlinear Schrodinger-Poisson equations, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 4, 745-764. https://doi.org/10.1017/S0308210511000692
  9. X. Chen and C. Tang, Least energy sign-changing solutions for Schrodinger-Poisson system with critical growth, Commun. Pure Appl. Anal. 20 (2021), no. 6, 2291-2312. https://doi.org/10.3934/cpaa.2021077
  10. S. Chen and X. Tang, Ground state sign-changing solutions for a class of SchrodingerPoisson type problems in ℝ3, Z. Angew. Math. Phys. 67 (2016), no. 4, Art. 102, 18 pp. https://doi.org/10.1007/s00033-016-0695-2
  11. S. Chen and X. Tang, Radial ground state sign-changing solutions for a class of asymptotically cubic or super-cubic Schrodinger-Poisson type problems, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM 113 (2019), no. 2, 627-643. https://doi.org/10.1007/s13398-018-0493-0
  12. T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 5, 893-906. https://doi.org/10.1017/S030821050000353X
  13. M. F. Furtado, L. A. Maia, and E. S. Medeiros, Positive and nodal solutions for a nonlinear Schrodinger equation with indefinite potential, Adv. Nonlinear Stud. 8 (2008), no. 2, 353-373. https://doi.org/10.1515/ans-2008-0207
  14. M. F. Furtado, L. A. Maia, and E. S. Medeiros, A note on the existence of a positive solution for a non-autonomous Schrodinger-Poisson system, in Analysis and topology in nonlinear differential equations, 277-286, Progr. Nonlinear Differential Equations Appl., 85, Birkhauser/Springer, Cham, 2014.
  15. L. Huang and E. M. Rocha, A positive solution of a Schrodinger-Poisson system with critical exponent, Commun. Math. Anal. 15 (2013), no. 1, 29-43.
  16. I. Ianni, Sign-changing radial solutions for the Schrodinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal. 41 (2013), no. 2, 365-385.
  17. S. Kim and J. Seok, On nodal solutions of the nonlinear Schrodinger-Poisson equations, Commun. Contemp. Math. 14 (2012), no. 6, 1250041, 16 pp. https://doi.org/10.1142/S0219199712500411
  18. Z. Liang, J. Xu, and X. Zhu, Revisit to sign-changing solutions for the nonlinear Schrodinger-Poisson system in ℝ3, J. Math. Anal. Appl. 435 (2016), no. 1, 783-799. https://doi.org/10.1016/j.jmaa.2015.10.076
  19. Z. Liu, Z.-Q. Wang, and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrodinger-Poisson system, Ann. Mat. Pura Appl. (4) 195 (2016), no. 3, 775-794. https://doi.org/10.1007/s10231-015-0489-8
  20. C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. (2) 3 (1940), 5-7.
  21. D. Ruiz, The Schrodinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), no. 2, 655-674. https://doi.org/10.1016/j.jfa.2006.04.005
  22. O. Sanchez and J. Soler, Long-time dynamics of the Schrodinger-Poisson-Slater system, J. Statist. Phys. 114 (2004), no. 1-2, 179-204. https://doi.org/10.1023/B:JOSS.0000003109.97208.53
  23. W. Shuai and Q. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrodinger-Poisson system in ℝ3, Z. Angew. Math. Phys. 66 (2015), no. 6, 3267-3282. https://doi.org/10.1007/s00033-015-0571-5
  24. G. Siciliano, Multiple positive solutions for a Schrodinger-Poisson-Slater system, J. Math. Anal. Appl. 365 (2010), no. 1, 288-299. https://doi.org/10.1016/j.jmaa.2009.10.061
  25. J. Sun, T. Wu, and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrodinger-Poisson system, J. Differential Equations 260 (2016), no. 1, 586-627. https://doi.org/10.1016/j.jde.2015.09.002
  26. D.-B. Wang, H.-B. Zhang, and W. Guan, Existence of least-energy sign-changing solutions for Schrodinger-Poisson system with critical growth, J. Math. Anal. Appl. 479 (2019), no. 2, 2284-2301. https://doi.org/10.1016/j.jmaa.2019.07.052
  27. Z. Wang and H.-S. Zhou, Sign-changing solutions for the nonlinear Schrodinger-Poisson system in ℝ3, Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 927-943. https://doi.org/10.1007/s00526-014-0738-5
  28. T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations 27 (2006), no. 4, 421-437. https://doi.org/10.1007/s00526-006-0015-3
  29. M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhauser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
  30. Y. Yu, F. Zhao, and L. Zhao, Positive and sign-changing least energy solutions for a fractional Schrodinger-Poisson system with critical exponent, Appl. Anal. 99 (2020), no. 13, 2229-2257. https://doi.org/10.1080/00036811.2018.1557325
  31. Z. Zhang, Y. Wang, and R. Yuan, Ground state sign-changing solution for Schrodinger-Poisson system with critical growth, Qual. Theory Dyn. Syst. 20 (2021), no. 2, Paper No. 48, 23 pp. https://doi.org/10.1007/s12346-021-00487-5