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Tytuł artykułu

A comprehensive review on the existence of normalized solutions for four classes of nonlinear elliptic equations

Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper provides a comprehensive review of recent results concerning the existence of normalized solutions for four classes of nonlinear elliptic equations: Schrödinger equations, Schrödinger–Poisson equations, Kirchhoff equations, and Choquard equations.
Rocznik
Strony
739--763
Opis fizyczny
Bibliogr. 53 poz.
Twórcy
autor
  • HNP-LAMA, Central South University, School of Mathematics and Statistics, Changsha, Hunan 410083, P.R. China
autor
  • HNP-LAMA, Central South University, School of Mathematics and Statistics, Changsha, Hunan 410083, P.R. China
Bibliografia
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  • [7] B. Bieganowski, J. Mederski, Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth, J. Funct. Anal. 280 (2021), no. 11, Paper no. 108989.
  • [8] I. Catto, P.L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas–Fermi type theories. Part I: a necessary and sufficient condition for the stability of general molecular systems, Commun. Partial Differ. Equ. 17 (1992), 1051–1110.
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  • [11] S.T. Chen, X.H. Tang, Axially symmetric solutions for the planar Schrödinger–Poisson system with critical exponential growth, J. Differ. Equ. 269 (2020), 9144–9174.
  • [12] S.T. Chen, X.H. Tang, Normalized solutions for Schrödinger equations with mixed dispersion and critical exponential growth in R2, Calc. Var. Partial Differ. Equations 62 (2023), Paper no. 261.
  • [13] S.T. Chen, X.H. Tang, Another look at Schrödinger equations with prescribed mass, J. Differential Equations 386 (2024), 435–479.
  • [14] S.T. Chen, X.H. Tang, Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities, Math. Ann. 391 (2025), 2783–2836.
  • [15] S.T. Chen, V.D. Radulescu, X.H. Tang, Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth, Math. Z. 306 (2024), Paper no. 50.
  • [16] S.T. Chen, J.P. Shi, X.H. 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.
  • [17] S.T. Chen, X.H. Tang, S. Yuan, Normalized solutions for Schrödinger–Poisson equations with general nonlinearities, J. Math. Anal. Appl. 481 (2020), 123447.
  • [18] S. Cingolani, L. Jeanjean, Stationary waves with prescribed L2-norm for the planar Schrödinger–Poisson system, SIAM J. Math. Anal. 51 (2019), 3533–3568.
  • [19] S. Cingolani, K. Tanaka, Ground State Solutions for the Nonlinear Choquard Equation with Prescribed Mass, Springer INdAM Ser., vol. 47, Springer, Cham, 2021.
  • [20] Q. Gao, X.M. He, Normalized solutions for the Choquard equations with critical nonlinearities, Adv. Nonlinear Anal. 13 (2024), 20240030.
  • [21] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, vol. 107 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993.
  • [22] Q.H. He, Z.Y. Lv, Y.M. Zhang, X.X. Zhong, Existence and blow up behavior of positive normalized solution to the Kirchhoff equation with general nonlinearities: Mass super-critical case, J. Differential Equations 356 (2023), 375–406.
  • [23] J. Hirata, K. Tanaka, Nonlinear scalar field equations with L2-constraint: Mountain pass and symmetric mountain pass approaches, Adv. Nonlinear Stud. 19 (2019), 263–290.
  • [24] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997), 1633–1659.
  • [25] L. Jeanjean, T.T. Le, Multiple normalized solutions for a Sobolev critical Schrödinger equation, Math. Ann. 384 (2021), 101–134.
  • [26] L. Jeanjean, S.S. Lu, Nonradial normalized solutions for nonlinear scalar field equations, Nonlinearity 32 (2019), 4942–4966.
  • [27] L. Jeanjean, S.S. Lu, A mass supercritical problem revisited, Calc. Var. 59 (2020), Article no. 174.
  • [28] L. Jeanjean, S.S. Lu, On global minimizers for a mass constrained problem, Calc. Var. 61 (2022), Article no. 214.
  • [29] L. Jeanjean, T. Luo, Sharp nonexistence results of prescribed L2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations, Z. Angew. Math. Phys. 64 (2013), 937–954.
  • [30] L. Jeanjean, J. Jendrej, T.T. Le, N. Visciglia, Orbital stability of ground states for a Sobolev critical Schrödinger equation, J. Math. Pures Appl. (9) 164 (2022), 158–179.
  • [31] M.K. Kwong, Uniqueness of positive solutions of −Δu+u = up−1 in Rn, Arch. Rational Mech. Anal. 105 (1989), 243–266.
  • [32] G. Li, X. Luo, T. Yang, Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent, Ann. Fenn. Math. 47 (2022), 895–925.
  • [33] G. Li, H. Ye, The existence of positive solutions with prescribed L2-norm for nonlinear Choquard equations, J. Math. Phys. 55 (2014), 1–19.
  • [34] X.F. Li, Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability, Adv. Nonlinear Anal. 11 (2022), 1134–1164.
  • [35] X. Luo, Q.F. Wang, Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in R3, Nonlinear Anal. Real World Appl. 33 (2017), 19–32.
  • [36] S.J. Qi, W.M. Zou, Exact number of the positive solutions for Kirchhoff equation, SIAM J. Math. Anal. 54 (2022), 5424–5446.
  • [37] O. Sanchez, J. Soler, Long-time dynamics of the Schrödinger–Poisson–Slater system, J. Stat. Phys. 114 (2004), 179–204.
  • [38] M. Shibata, Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term, Manuscripta Math. 143 (2014), 221–37.
  • [39] N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differ. Equ. 269 (2020), 6941–6987.
  • [40] N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), 108610.
  • [41] C.A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. Lond. Math. Soc. 45 (1982), 169–192.
  • [42] C.A. Stuart, Bifurcation from the essential spectrum for some non-compact non-linearities, Math. Methods Appl. Sci. 11 (1989), 525–542.
  • [43] Q. Wang, A.X. Qian, Normalized solutions to the Schrödinger–Poisson-Slater equation with general nonlinearity: mass supercritical case, Anal. Math. Phys. 13 (2023), Article no. 35.
  • [44] Q. Wang, A.X. Qian, Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case, J. Inequal. Appl. (2024), Article no. 48.
  • [45] J. Wei, Y. Wu, Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal. 283 (2022), Paper no. 109574.
  • [46] W.H. Xie, H.B. Chen, H.X. Shi, Existence and multiplicity of normalized solutions for a class of Schrödinger–Poisson equations with general nonlinearities, Math. Meth. Appl. Sci. 43 (2020), 3602–3616.
  • [47] H. Ye, Mass minimizers and concentration for nonlinear Choquard equations in RN, Topol. Methods Nonlinear Anal. 48 (2016), 393–417.
  • [48] H.Y. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci. 38 (2015), 2663–2679.
  • [49] H.Y. Ye, The existence of normalized solutions for L2-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys. 66 (2015), 1483–1497.
  • [50] H.Y. Ye, The mass concentration phenomenon for L2-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys. 67 (2016), Article no. 29.
  • [51] P. Zhang, Z. Han, Normalized ground states for Kirchhoff equations in R3 with a critical nonlinearity, J. Math. Phys. 63 (2022), Paper no. 021505.
  • [52] Y.L. Zeng, K.S. Chen, Remarks on normalized solutions for L2-critical Kirchhoff problems, Taiwan. J. Math. 20 (2016), 617–627.
  • [53] X.Y. Zeng, Y.M. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett. 74 (2017) 52–59.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025)
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-f09e5805-a0c8-413b-8625-f1477bbf8b98
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