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Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

  • Nils Ackermann
  • Alfredo Cano
  • Eric Hernández-Martínez
Article

Abstract

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.

Keywords

Weak Solution Homogeneous Boundary Condition Cylindrical Domain Smooth Bounded Domain Critical Sobolev Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank the anonymous referee for carefully reading the manuscript and for suggesting several improvements.

References

  1. 1.
    Bahri, A., Lions, P.L.: Morse index of some min-max critical points. I. Application to multiplicity results. Commun. Pure Appl. Math. 41(8), 1027–1037 (1988). doi: 10.1002/cpa.3160410803
  2. 2.
    Tanaka, K.: Morse indices at critical points related to the symmetric mountain pass theorem and applications. Commun. Partial Differ. Equ. 14(1), 99–128 (1989). doi: 10.1080/03605308908820592
  3. 3.
    Bahri, A., Berestycki, H.: A perturbation method in critical point theory and applications. Trans. Am. Math. Soc. 267(1), 1–32 (1981). doi: 10.2307/1998565
  4. 4.
    Rabinowitz, P.H.: Multiple critical points of perturbed symmetric functionals. Trans. Am. Math. Soc. 272(2), 753–769 (1982). doi: 10.2307/1998726 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Struwe, M.: Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscr. Math. 32(3–4), 335–364 (1980). doi: 10.1007/BF01299609 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bahri, A.: Topological results on a certain class of functionals and application. J. Funct. Anal. 41(3), 397–427 (1981). doi: 10.1016/0022-1236(81)90083-5 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bartolo, R., Candela, A.M., Salvatore, A.: Infinitely many solutions for a perturbed Schrödinger equation. Discrete Contin. Dyn. Syst. (Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl.), 94–102 (2015). doi: 10.3934/proc.2015.0094
  8. 8.
    Hirano, N., Zou, W.: A perturbation method for multiple sign-changing solutions. Calc. Var. Partial Differ. Equ. 37(1–2), 87–98 (2010). doi: 10.1007/s00526-009-0253-2 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Li, Y., Liu, Z., Zhao, C.: Nodal solutions of a perturbed elliptic problem. Topol. Methods Nonlinear Anal. 32(1), 49–68 (2008)MathSciNetMATHGoogle Scholar
  10. 10.
    Li, S., Liu, Z.: Perturbations from symmetric elliptic boundary value problems. J. Differ. Equ. 185(1), 271–280 (2002). doi: 10.1006/jdeq.2001.4160 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Magrone, P., Mataloni, S.: Multiple solutions for perturbed indefinite semilinear elliptic equations. Adv. Differ. Equ. 8(9), 1107–1124 (2003)MathSciNetMATHGoogle Scholar
  12. 12.
    Tarsi, C.: Perturbation from symmetry and multiplicity of solutions for elliptic problems with subcritical exponential growth in \(\mathbb{R}^2\). Commun. Pure Appl. Anal. 7(2), 445–456 (2008). doi: 10.3934/cpaa.2008.7.445 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Tehrani, H.T.: Infinitely many solutions for indefinite semilinear elliptic equations without symmetry. Commun. Partial Differ. Equ. 21(3–4), 541–557 (1996)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Li, Y.Y.: Existence of infinitely many critical values of some nonsymmetric functionals. J. Differ. Equ. 95(1), 140–153 (1992). doi: 10.1016/0022-0396(92)90046-P MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Salvatore, A.: Multiple solutions for perturbed elliptic equations in unbounded domains. Adv. Nonlinear Stud. 3(1), 1–23 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bolle, P., Ghoussoub, N., Tehrani, H.: The multiplicity of solutions in non-homogeneous boundary value problems. Manuscr. Math. 101(3), 325–350 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Candela, A.M., Salvatore, A.: Multiplicity results of an elliptic equation with non-homogeneous boundary conditions. Topol. Methods Nonlinear Anal. 11(1), 1–18 (1998)MathSciNetMATHGoogle Scholar
  18. 18.
    Struwe, M.: Superlinear elliptic boundary value problems with rotational symmetry. Arch. Math. (Basel) 39(3), 233–240 (1982). doi: 10.1007/BF01899529 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Castro, A., Kurepa, A.: Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball. Proc. Am. Math. Soc. 101(1), 57–64 (1987). doi: 10.2307/2046550 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kazdan, J.L., Warner, F.W.: Remarks on some quasilinear elliptic equations. Commun. Pure Appl. Math. 28(5), 567–597 (1975)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Candela, A.M., Palmieri, G., Salvatore, A.: Radial solutions of semilinear elliptic equations with broken symmetry. Topol. Methods Nonlinear Anal. 27(1), 117–132 (2006)MathSciNetMATHGoogle Scholar
  22. 22.
    Barile, S., Salvatore, A.: Multiplicity results for some perturbed elliptic problems in unbounded domains with non-homogeneous boundary conditions. Nonlinear Anal. 110, 47–60 (2014). doi: 10.1016/j.na.2014.07.018 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Barile, S., Salvatore, A.: Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Discrete Contin. Dyn. Syst. (Dynamical systems, differential equations and applications. 9th AIMS Conference. Suppl.), 41–49 (2013). doi: 10.3934/proc.2013.2013.41
  24. 24.
    Clapp, M., Hernández-Martínez, E.: Infinitely many solutions of elliptic problems with perturbed symmetries in symmetric domains. Adv. Nonlinear Stud. 6(2), 309–322 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Brüning, J., Heintze, E.: Representations of compact Lie groups and elliptic operators. Invent. Math. 50(2), 169–203 (1978/1979). doi: 10.1007/BF01390288
  26. 26.
    Donnelly, H.: \(G\)-spaces, the asymptotic splitting of \(L^{2}(M)\) into irreducibles. Math. Ann. 237(1), 23–40 (1978). doi: 10.1007/BF01351556
  27. 27.
    Barile, S., Salvatore, A.: Multiplicity results for some perturbed and unperturbed “zero mass” elliptic problems in unbounded cylinders. In: Analysis and Topology in Nonlinear Differential Equations, Progr. Nonlinear Differential Equations Appl., vol. 85, pp. 39–59. Birkhäuser/Springer, Cham (2014)Google Scholar
  28. 28.
    Bolle, P.: On the Bolza problem. J. Differ. Equ. 152(2), 274–288 (1999)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88(3), 309–318 (1983). http://projecteuclid.org/getRecord?id=euclid.cmp/1103922378
  30. 30.
    Blanchard, P., Stubbe, J., Rezende, J.: New estimates on the number of bound states of Schrödinger operators. Lett. Math. Phys. 14(3), 215–225 (1987). doi: 10.1007/BF00416851
  31. 31.
    Hebey, E., Vaugon, M.: Sobolev spaces in the presence of symmetries. J. Math. Pures Appl. (9) 76(10), 859–881 (1997). doi: 10.1016/S0021-7824(97)89975-8 MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wang, W.: Sobolev embeddings involving symmetry. Bull. Sci. Math. 130(4), 269–278 (2006). doi: 10.1016/j.bulsci.2005.05.004 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lieb, E.: Bounds on the eigenvalues of the Laplace and Schroedinger operators. Bull. Am. Math. Soc. 82(5), 751–753 (1976)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. (2) 106(1), 93–100 (1977)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Rosenbljum, G.: Distribution of the discrete spectrum of singular differential operators. Sov. Math. Dokl. 13, 245–249 (1972)Google Scholar
  36. 36.
    Bargmann, V.: On the number of bound states in a central field of force. Proc. Natl. Acad. Sci. USA 38, 961–966 (1952)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Schmidt, K.M.: A short proof for Bargmann-type inequalities. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458(2027), 2829–2832 (2002). doi: 10.1098/rspa.2002.1021 MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Clapp, M., Hernández-Linares, S., Hernández-Martínez, E.: Linking-preserving perturbations of symmetric functionals. J. Differ. Equ. 185(1), 181–199 (2002). doi: 10.1006/jdeq.2002.4170
  39. 39.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (1986)Google Scholar
  40. 40.
    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc., New York (1991)Google Scholar
  41. 41.
    Davies, E.B.: Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989). doi: 10.1017/CBO9780511566158
  42. 42.
    König, H.: Eigenvalue distribution of compact operators, Operator Theory: Advances and Applications, vol. 16. Birkhäuser Verlag, Basel (1986). doi: 10.1007/978-3-0348-6278-3
  43. 43.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, 2nd edn. Springer, Berlin (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Nils Ackermann
    • 1
  • Alfredo Cano
    • 2
  • Eric Hernández-Martínez
    • 3
  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico, D.F.Mexico
  2. 2.Facultad de CienciasUniversidad Autónoma del Estado de MéxicoTolucaMexico
  3. 3.Universidad Autónoma de la Ciudad de MéxicoMexico, D.F.Mexico

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