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Bifurcation from the essential spectrum for an elliptic equation with general nonlinearities

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Abstract

In this paper, based on some prior estimates, we show that the essential spectrum λ = 0 is a bifurcation point for a superlinear elliptic equation with only local conditions, which generalizes a series of earlier results on an open problem proposed by Stuart (1983).

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References

  1. Armstrong S N, Sirakov B. Nonexistence of positive supersolutions of elliptic equations via the maximum principle. Comm Partial Differential Equations, 2011, 36: 2011–2047

    Article  MathSciNet  MATH  Google Scholar 

  2. Benci V, Fortunato D. Does bifurcation from the essential spectrum occur? Comm Partial Differential Equations, 1981, 6: 249–272

    Article  MathSciNet  MATH  Google Scholar 

  3. Benci V, Fortunato D. Bifurcation from the essential spectrum for odd variational operators. Confer Sem Mat Univ Bari, 1981, 178: 26pp

    MathSciNet  MATH  Google Scholar 

  4. Berestycki H, Lions P L. Existence of stationary states in nonlinear scalar field equations. In: Bifurcation Phenomena in Mathematical Physics and Related Topics. Dordrecht-Boston-London: D. Reidel, 1976, 269–292

  5. Berestycki H, Lions P L. Nonlinear scalar field equations, I: Existence of a ground state. Arch Ration Mech Anal, 1983, 82: 313–345

    Article  MathSciNet  MATH  Google Scholar 

  6. Bongers A, Heinz H P, Küpper T. Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains. J Differential Equations, 1983, 47: 327–357

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen S W, Liu Z L, Wang Z Q. A variant of Clark’s theorem and its applications for nonsmooth functionals without the Palais-Smale condition. SIAM J Math Anal, 2017, 49: 446–470

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen W X, Li C M. Classification of solutions of some nonlinear elliptic equations. Duke Math J, 1991, 63: 615–622

    Article  MathSciNet  MATH  Google Scholar 

  9. Clark D C. A variant of the Lusternik-Schnirelman theory. Indiana Univ Math J, 1972, 22: 65–74

    Article  MathSciNet  MATH  Google Scholar 

  10. Dautray R, Lions J L. Analyse mathématique et calcul numérique pour les sciences et les techniques. Paris: Masson, 1984

    MATH  Google Scholar 

  11. Gidas B, Spruck J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm Pure Appl Math, 1981, 34: 525–598

    Article  MathSciNet  MATH  Google Scholar 

  12. Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 2001

    Book  MATH  Google Scholar 

  13. Guo Y J, Seiringer R. On the mass concentration for Bose-Einstein condensates with attractive interactions. Lett Math Phys, 2014, 104: 141–156

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo Y J, Zeng X Y, Zhou H S. Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials. Ann Inst H Poincaré Anal Non Linéaire, 2016, 33: 809–828

    Article  MathSciNet  MATH  Google Scholar 

  15. He X M, Zou W M. Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth. Sci China Math, 2020, 63: 1571–1612

    Article  MathSciNet  MATH  Google Scholar 

  16. Heinz H P. Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm-Liouville problems. J Differential Equations, 1986, 64: 79–108

    Article  MathSciNet  MATH  Google Scholar 

  17. Heinz H P. Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems. J Differential Equations, 1987, 66: 263–300

    Article  MathSciNet  MATH  Google Scholar 

  18. Jeanjean L. Local conditions insuring bifurcation from the continuous spectrum. Math Z, 1999, 232: 651–664

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeanjean L, Tanaka K. A remark on least energy solutions in ℝN. Proc Amer Math Soc, 2002, 131: 2399–2408

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeanjean L, Tanaka K. A note on a mountain pass characterization of least energy solutions. Adv Nonlinear Stud, 2003, 3: 445–455

    Article  MathSciNet  MATH  Google Scholar 

  21. Jeanjean L, Zhang J J, Zhong X X. A global branch approach to normalized solutions for the Schrödinger equation. arXiv:2112.05869, 2021

  22. Li H W, Yang Z, Zou W M. Normalized solutions for nonlinear Schrödinger equations (in Chinese). Sci Sin Math, 2020, 50: 1023–1044

    Article  MATH  Google Scholar 

  23. Liu Z L, Wang Z Q. On Clark’s theorem and its applications to partially sublinear problems. Ann Inst H Poincaré Anal Non Linéaire, 2015, 32: 1015–1037

    Article  MathSciNet  MATH  Google Scholar 

  24. Quittner P, Souplet P. Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States. Basel: Birkhäuser, 2007

    MATH  Google Scholar 

  25. Shibata M. Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term. Manuscripta Math, 2014, 143: 221–237

    Article  MathSciNet  MATH  Google Scholar 

  26. Stuart C A. Bifurcation for variational problems when the linearisation has no eigenvalues. J Funct Anal, 1980, 38: 169–187

    Article  MathSciNet  MATH  Google Scholar 

  27. Stuart C A. Bifurcation from the continuous spectrum in the L2-theory of elliptic equations on ℝN. In: Recent Methods in Nonlinear Analysis and Applications. Napoli: Liguori, 1981, 231–300

    Google Scholar 

  28. Stuart C A. Bifurcation for Dirichlet problems without eigenvalues. Proc Lond Math Soc (3), 1982, 45: 169–192

    Article  MathSciNet  MATH  Google Scholar 

  29. Stuart C A. Bifurcation from the essential spectrum. In: Lecture Notes in Mathematics, vol. 1017. Berlin: Springer, 1983, 575–596

    MATH  Google Scholar 

  30. Stuart C A. Bifurcation from the essential spectrum for some noncompact nonlinearities. Math Methods Appl Sci, 1989, 11: 525–542

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang Z Q. Nonlinear boundary value problems with concave nonlinearities near the origin. NoDEA Nonlinear Differential Equations Appl, 2001, 8: 15–33

    Article  MathSciNet  MATH  Google Scholar 

  32. Willem M. Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Boston: Birkhäuser, 1996

    Book  MATH  Google Scholar 

  33. Yang J F, Yang J G. Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations. Sci China Math, 2022, 65: 1383–1412

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhu X P, Zhou H S. Bifurcation from the essential spectrum of superlinear elliptic equations. Appl Anal, 1988, 28: 51–66

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11801581, 11871123, 11931012 and 12271184), Guangdong Province Science Foundation (Grant Nos. 2021A1515010034 and 2018A030310082), Guangzhou Science Foundation (Grant No. 202102020225), Chongqing Science Foundation (Grant No. JDDSTD201802) and Chongqing University Science Foundation (Grant No. CXQT21021). This work was partially carried out when the second author visited the Center for Mathematical Sciences (CMS) at Wuhan University of Technology (WUT), and he thanks CMS-WUT for its hospitality.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Jianjun Zhang

  2. South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, 510631, China

    Xuexiu Zhong

  3. Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan, 430070, China

    Huansong Zhou

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  1. Jianjun Zhang
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  2. Xuexiu Zhong
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  3. Huansong Zhou
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Correspondence to Huansong Zhou.

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Zhang, J., Zhong, X. & Zhou, H. Bifurcation from the essential spectrum for an elliptic equation with general nonlinearities. Sci. China Math. 66, 2243–2260 (2023). https://doi.org/10.1007/s11425-022-2049-1

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