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Normalized solutions to the Schrödinger–Poisson–Slater equation with general nonlinearity: mass supercritical case

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Abstract

We study the following nonlinear mass supercritical Schrödinger–Poisson–Slater equation

$$\begin{aligned} -\triangle u+\mu u-\gamma (|x|^{-1}*|u|^2)u-af(u)=0 \ \ \text {in} \ {{\mathbb {R}}^{3}} \end{aligned}$$

satisfying the normalized constrain \(\int _{{\mathbb {R}}^{3}}|u|^{2}dx=m\), and \(m>0\) prescribed. The nonlinearity f is super-critical. Under some mild assumptions, we establish the existence of ground state and infinitely many radial solutions by constructing a particular bounded Palais-Smale sequence when \(\gamma <0,a>0\). Meanwhile, we obtain the non-existence result in the case \(\gamma<0,a<0\) and the existence result when \(\gamma >0,a<0\) via variational methods.

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References

  1. Sánchez, O., Soler, J.: Long-time dynamics of the Schrödinger-Poisson-Slater system. J. Stat. Phys. 114, 179–204 (2004)

    Article  MATH  Google Scholar 

  2. Kikuchi, H.: Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 7, 403–437 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellazzini, J., Siciliano, G.: Stable standing waves for a class of nonlinear Schrödinger-Poisson equations. Z. Angew. Math. Phys. 62, 267–280 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellazzini, J., Siciliano, G.: Scaling properties of functionals and existence of constrained minimizers. J. Funct. Anal. 261, 2486–2507 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jeanjean, L., Luo, T.J.: Sharp nonexistence results of prescribed \(L^2\) -norm solutions for some class of Schrödinger Poisson and quasilinear equations. Z. Angrew. Math. Phys. 64, 937–954 (2013)

    Article  MATH  Google Scholar 

  6. Bellazzini, J., Jeanjean, L., Luo, T.J.: Existence and instability of standing waves with prescribed norm for a class of Schrödinger Poisson equations. Proc. Lond. Math. Soc. 107, 303–339 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bartsch, T., De Valeriola, S.: Normalized solutions of nonlinear Schrödinger equations. Arch. Math. 100, 75–83 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Luo, T.J.: Multiplicity of normalized solutions for a class of nonlinear Schrödinger-Poisson-Slater equations. J. Math. Anal. Appl. 416, 195–204 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ye, H.Y.: The existence and the concentration behavior of normalized solutions for the \(L^2\)-critical Schrödinger-Poisson system. Comput. Math. Appl. 74, 266–280 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Ye, H.Y., Luo, T.J.: On the mass concentration of \(L^2\)-constrained minimizers for a class of Schrödinger-Poisson equations. Z Angew. Math. Phys. 69, 66 (2018)

    Article  MATH  Google Scholar 

  11. Ye, H.Y., Zhang, L.: Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials. J. Math. Anal. Appl. 452, 47–61 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MATH  Google Scholar 

  13. Alves, A. O., Böer, E. de S., Miyagaki, O. H.: Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity. arXiv e-prints (2021). https://doi.org/10.48550/arXiv.2107.13281,22

  14. Jeanjean, L., Le, T.T.: Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation. J. Differ. Equ. 303, 277–325 (2021)

    Article  MATH  Google Scholar 

  15. Jeanjean, L., Lu, S.S.: A mass supercritical problem revisited. Calc. Var. 59, 174 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bieganowski, B., Mederski, J.: Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth. J. Funct. Anal. 280, 11 (2021)

    Article  MATH  Google Scholar 

  17. Ikoma, N., Miyamoto, Y.: Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities. Calc. Var. 59, 48 (2020)

    Article  MATH  Google Scholar 

  18. Xie, W. H., Chen, H. B., Shi, H. X.: Existence and multiplicity of normalized solutions for a class of Schrödinger-Poisson equations with general nonlinearities. Mathematical Methods in the Applied Sciences, 2020, 43(6)

  19. Chen, S.T., Tang, X.H., Yuan, S.: Normalized solutions for Schrödinger-Poisson equations with general nonlinearities. J. Math. Anal. Appl. 481, 123447 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28, 1633–1659 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bartsch, T., Soave, N.: Correction to A natural constraint approach to normalized solutions on nonlinear Schrödinger equations and systems . [J. Funct. Anal. 272(2017)4998–5037]. J. Funct. Anal. 275(2018) 516–521

  22. Bartsch, T., Soave, N.: Multiple normalized solutions for a competting system of Schrödinger equations. Calc. Var. 58(22)(2019)

  23. Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhao, L.G., Zhang, F.K.: On the existence of solutions for the Schrödinger Poisson equations. J. Math. Anal. Appl. 346, 155–69 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  MATH  Google Scholar 

  26. Tang, X.H., Chen, S.T.: Ground state solutions of Nehari Poho\(\breve{\text{ z }}\)aev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56, 110–134 (2017)

    Article  Google Scholar 

  27. Lions, P. L. : The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 223–283 (1984)

  28. Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257, 3802–3822 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Szulkin, A., Weth, T.: The Method of Nehari Manifold. Handbook of Nonconvex Analysis and Applica-tions. (2010)

  30. Chang, K. C.: Methods in Nonlinear Analysis, Springer Monographs in Mathematics. (2005)

  31. Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

  32. Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  33. Berestycki, H., Lions, P.L.: Nonlinear scalar field equations II: existence of infinitely many solutions. Arch. Rat. Mech. Anal. 82, 347–375 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Strauss, W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  35. Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I: existence of a ground state. Arch. Rat. Mech. Anal. 82, 313–346 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics. (1986)

  37. KrasnoselskI, M.A.: Topological Methods in the Theory of Nonlinear Integra Equations. Macmillan, New York (1964)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, People’s Republic of China

    Qun Wang & Aixia Qian

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  1. Qun Wang
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  2. Aixia Qian
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All authors contributed equally and significantly in writing this article. All authors wrote, read, and approved the final manuscript.

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Correspondence to Aixia Qian.

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Supported by the SNSFC(ZR2021MA096, ZR2020MA005).

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Wang, Q., Qian, A. Normalized solutions to the Schrödinger–Poisson–Slater equation with general nonlinearity: mass supercritical case. Anal.Math.Phys. 13, 35 (2023). https://doi.org/10.1007/s13324-023-00788-9

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  • DOI: https://doi.org/10.1007/s13324-023-00788-9

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