Skip to main content
SpringerLink
Log in

Normalized solutions for a biharmonic Choquard equation with exponential critical growth in \(\mathbb {R}^4\)

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we study the following biharmonic Choquard-type problem

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} \Delta ^2u-\beta \Delta u=\lambda u+(I_\mu *F(u))f(u), \quad \text{ in }\ \ \mathbb {R}^4,\\ \displaystyle \int \limits _{\mathbb {R}^4}|u|^2\textrm{d}x=c^2>0,\quad u\in H^2(\mathbb {R}^4), \end{array} \right. \end{aligned} \end{aligned}$$

where \(\beta \ge 0\), \(\lambda \in \mathbb {R}\), \(I_\mu =\frac{1}{|x|^\mu }\) with \(\mu \in (0,4)\), F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Alves, C.O., Ji, C., Miyagaki, O.H.: Normalized solutions for a Schrödinger equation with critical growth in \(\mathbb{R} ^N\). Calc. Var. Partial Differ. Equ. 61, 18 (2022)

    Google Scholar 

  2. Bartsch, T., Liu, Y.Y., Liu, Z.L.: Normalized solutions for a class of nonlinear Choquard equations. Partial Differ. Equ. Appl. 1, 34 (2020)

    MathSciNet  Google Scholar 

  3. Bartsch, T., Soave, N.: A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. J. Funct. Anal. 272, 4998–5037 (2017)

    MathSciNet  Google Scholar 

  4. Bartsch, T., Soave, N.: Multiple normalized solutions for a competing system of Schrödinger equations. Calc. Var. Partial Differ. Equ. 58, 22 (2019)

    Google Scholar 

  5. Battaglia, L., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations in the plane. Adv. Nonlinear Stud. 17, 581–594 (2017)

    MathSciNet  Google Scholar 

  6. Bieganowski, B., Mederski, J.: Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth. J. Funct. Anal. 280, 108989 (2021)

    Google Scholar 

  7. Bonheure, D., Casteras, J.-B., dos Santos, E., Nascimento, R.: Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J. Math. Anal. 50, 5027–5071 (2018)

    MathSciNet  Google Scholar 

  8. Bonheure, D., Casteras, J.-B., Gou, T.X., Jeanjean, L.: Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime. Trans. Am. Math. Soc. 372, 2167–2212 (2019)

    Google Scholar 

  9. Boussaïd, N., Fernández, A.J., Jeanjean, L.: Some remarks on a minimization problem associated to a fourth order nonlinear Schrödinger equation. arXiv preprint (2019). arXiv:1910.13177

  10. Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)

    Google Scholar 

  11. Chen, J.Q., Chen, Z.W.: Multiple normalized solutions for biharmonic Choquard equation with Hardy–Littlewood–Sobolev upper critical and combined nonlinearities. J. Geom. Anal. 33, 371 (2023)

    MathSciNet  Google Scholar 

  12. Chen, W.J., Wang, Z.X.: Normalized ground states for a biharmonic Choquard equation with exponential critical growth. arXiv preprint (2022). arXiv:2211.13701

  13. Cingolani, S., Gallo, M., Tanaka, K.: Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities. Calc. Var. Partial Differ. Equ. 61, 68 (2022)

    MathSciNet  Google Scholar 

  14. Fernández, A.J., Jeanjean, L., Mandel, R., Mariş, M.: Non-homogeneous Gagliardo–Nirenberg inequalities in \(\mathbb{R} ^N\) and application to a biharmonic non-linear Schrödinger equation. J. Differ. Equ. 330, 1–65 (2022)

    Google Scholar 

  15. Fibich, G., Ilan, B., Papanicolaou, G.: Self-focusing with fourth-order dispersion. SIAM J. Appl. Math. 62, 1437–1462 (2002)

    MathSciNet  Google Scholar 

  16. Ghimenti, M., Van Schaftingen, J.: Nodal solutions for the Choquard equation. J. Funct. Anal. 271, 107–135 (2016)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  18. Jeanjean, L., Lu, S.S.: A mass supercritical problem revisited. Calc. Var. Partial Differ. Equ. 59, 174 (2020)

    MathSciNet  Google Scholar 

  19. Karpman, V.I.: Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations. Phys. Rev. E. 53, 1336–1339 (1996)

    Google Scholar 

  20. Karpman, V.I., Shagalov, A.G.: Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion. Phys. D 144, 194–210 (2000)

    MathSciNet  Google Scholar 

  21. Kavian, O.: Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Springer, Berlin (1993). (ISBN: 2-287-00410-6)

    Google Scholar 

  22. Li, X.F.: Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities. Calc. Var. Partial Differ. Equ. 60, 169 (2021)

    Google Scholar 

  23. Li, X.F.: Standing waves to upper critical Choquard equation with a local perturbation: multiplicity, qualitative properties and stability. Adv. Nonlinear Anal. 11, 1134–1164 (2022)

    MathSciNet  Google Scholar 

  24. Li, G.B., Ye, H.Y.: The existence of positive solutions with prescribed \(L^2\)-norm for nonlinear Choquard equations. J. Math. Phys. 55, 121501 (2014)

    MathSciNet  Google Scholar 

  25. Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)

    Google Scholar 

  26. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, II. Ann. l’Inst. Henri Poincaré Anal. Non Linéaire 1, 223–283 (1984)

    MathSciNet  Google Scholar 

  27. Lu, G.Z., Yang, Y.Y.: Adams’ inequalities for bi-Laplacian and extremal functions in dimension four. Adv. Math. 220, 1135–1170 (2009)

    MathSciNet  Google Scholar 

  28. Luo, X., Yang, T.: Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient. Sci. China Math. 66, 1237–1262 (2023)

    MathSciNet  Google Scholar 

  29. Luo, H.J., Zhang, Z.T.: Existence and stability of normalized solutions to the mixed dispersion nonlinear Schrödinger equations. Electron. Res. Arch. 30, 2871–2898 (2022)

    MathSciNet  Google Scholar 

  30. Luo, T.J., Zheng, S.J., Zhu, S.H.: The existence and stability of normalized solutions for a bi-harmonic nonlinear Schrödinger equation with mixed dispersion. Acta Math. Sci. Ser. B 43, 539–563 (2023)

    MathSciNet  Google Scholar 

  31. Miyagaki, O.H., Santana, C.R., Vieira, R.S.: Schrödinger equations in \(\mathbb{R} ^4\) involving the biharmonic operator with critical exponential growth. Rocky Mt. J. Math. 51, 243–263 (2021)

    Google Scholar 

  32. Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)

    MathSciNet  Google Scholar 

  33. Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)

    MathSciNet  Google Scholar 

  34. Moroz, V., Van Schaftingen, J.: A guide to the Choquard equation. J. Fixed Point Theory Appl. 19, 773–813 (2017)

    MathSciNet  Google Scholar 

  35. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13, 115–162 (1959)

    MathSciNet  Google Scholar 

  36. Noris, B., Tavares, H., Verzini, G.: Existence and orbital stability of the ground states with prescribed mass for the \(L^2\)-critical and supercritical NLS on bounded domains. Anal. PDE 7, 1807–1838 (2014)

    MathSciNet  Google Scholar 

  37. Pierotti, D., Verzini, G.: Normalized bound states for the nonlinear Schrödinger equation in bounded domains. Calc. Var. Partial Differ. Equ. 56, 133 (2017)

    Google Scholar 

  38. Ruf, B., Sani, F.: Sharp Adams-type inequalities in \(\mathbb{R} ^N\). Trans. Am. Math. Soc. 365, 645–670 (2013)

    Google Scholar 

  39. Ruiz, D., Van Schaftingen, J.: Odd symmetry of least energy nodal solutions for the Choquard equation. J. Differ. Equ. 264, 1231–1262 (2018)

    MathSciNet  Google Scholar 

  40. Shibata, M.: Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term. Manuscr. Math. 143, 221–237 (2014)

    Google Scholar 

  41. Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities. J. Differ. Equ. 269, 6941–6987 (2020)

    MathSciNet  Google Scholar 

  42. Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. J. Funct. Anal. 279, 108610 (2020)

    MathSciNet  Google Scholar 

  43. Stuart, C.A.: Bifurcation from the continuous spectrum in the \(L^2\)-theory of elliptic equations on \(\mathbb{R} ^N\). Recent methods in nonlinear analysis and applications, Liguori, Naples (1981)

  44. Stuart, C.A.: Bifurcation for Dirichlet problems without eigenvalues. Proc. Lond. Math. Soc. 45, 169–192 (1982)

    MathSciNet  Google Scholar 

  45. Wei, J.C., Wu, Y.Z.: Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities. J. Funct. Anal. 283, 109574 (2022)

    Google Scholar 

  46. Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Bosten Inc, Boston (1996). (ISBN: 0-8176-3913-9)

    Google Scholar 

  47. Yang, Y.Y.: Adams type inequalities and related elliptic partial differential equations in dimension four. J. Differ. Equ. 252, 2266–2295 (2012)

    MathSciNet  Google Scholar 

  48. Yao, S., Chen, H.B., Rădulescu, V., Sun, J.T.: Normalized solutions for lower critical Choquard equations with critical Sobolev perturbation. SIAM J. Math. Anal. 54, 3696–3723 (2022)

    MathSciNet  Google Scholar 

  49. Yuan, S., Chen, S.T., Tang, X.H.: Normalized solutions for Choquard equations with general nonlinearities. Electron. Res. Arch. 28, 291–309 (2020)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors have been supported by National Natural Science Foundation of China 11971392 and Natural Science Foundation of Chongqing, China cstc2021ycjh-bgzxm0115.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Wenjing Chen & Zexi Wang

Authors
  1. Wenjing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zexi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenjing Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Wang, Z. Normalized solutions for a biharmonic Choquard equation with exponential critical growth in \(\mathbb {R}^4\). Z. Angew. Math. Phys. 75, 58 (2024). https://doi.org/10.1007/s00033-024-02200-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-024-02200-3

Keywords

Mathematics Subject Classification

Search

Navigation