Multiplicity of Solutions to the Quasilinear Neumann Problem in the 3-Dimensional Case

We consider the quasilinear Neumann problem for equation with p-Laplacian in expanding three-dimensional balls. We prove that the number of essentially different positive solutions uboundedly increases with growth of radius.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    C. V. Coffman, “A non-linear boundary value problem with many positive solutions,” J. Differ. Equations 54, 429–437 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Y. Y. Li, “Existence of many positive solutions of semilinear elliptic equations on annulus,” J. Differ. Equations 83, 348–367 (1990).

    Article  MATH  Google Scholar 

  3. 3.

    J. Byeon, “Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli,” J. Differ. Equations 136, 136–165 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    A. I. Nazarov, “On solutions to the Dirichlet problem for an equation with p-Laplacian in a spherical layer” [in Russian], Tr. S. Peterb. Mat. O-va 10, 33-62 (2004); English transl.: Proc. St. Petersb. Math. Soc. 10, 29–57 (2005).

    Google Scholar 

  5. 5.

    S. B. Kolonitskii, “Multiplicity of solutions of the Dirichlet problem for an equation with the p-Laplacian in a three-dimensional spherical layer” [in Russian], Algebra Anal. 22, No. 3, 206–221 (2010); English transl.: St. Petersbg. Math. J. 22, No. 3, 485–495 (2011).

    Article  MathSciNet  Google Scholar 

  6. 6.

    A. P. Shcheglova, “Multiplicity of solutions to a boundary value problem with nonlinear Neumann condition” [in Russian], Probl. Mat. Anal. 30, 121–144 (2005); English transl.: J. Math. Sci., New York 128, No. 5, 3306–3333 (2005).

    Article  MathSciNet  Google Scholar 

  7. 7.

    N. Mizoguchi and T. Suzuki, “Semilinear elliptic equations on annulus for three and other space dimensions,” Houston J. Math. 1, 199–215 (1996).

    MathSciNet  Google Scholar 

  8. 8.

    P. L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case,” Ann. Inst. Henry Poincaré, Anal. Non Linéaire 1, 109–145; 223–283 (1984).

  9. 9.

    S. B. Kolonitskii, Multiplicity of 1D-concentrating positive solutions to the Dirichlet problem for equation with p-Laplacian, arXiv:1206.0787

  10. 10.

    B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer, Berlin (1985).

    Google Scholar 

  11. 11.

    R. S. Palais, “The principle of symmetric criticality,” Commun. Math. Phys. 69, 19–30 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    N. Trudinger, “On Harnack type inequalities and their application to quasilinear elliptic problems,” Commun. Pure Appl. Math. 20, 721–747 (1967).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. I. Nazarov.

Additional information

To Nina Nikolaevna, our Teacher

Translated from Problemy Matematicheskogo Analiza 78, January 2015, pp. 85-94.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Enin, A.I., Nazarov, A.I. Multiplicity of Solutions to the Quasilinear Neumann Problem in the 3-Dimensional Case. J Math Sci 207, 206–217 (2015). https://doi.org/10.1007/s10958-015-2366-9

Download citation

Keywords

  • Dirichlet Problem
  • Steklov Institute
  • Spherical Layer
  • Concentration Sequence
  • Unique Concentration