Skip to main content
SpringerLink
Log in

Existence of infinitely many solutions of nonlinear Steklov–Neumann problem

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study the existence of infinitely many solutions of the nonlinear Steklov–Neumann problem involving concave-convex type nonlinearities. The method of proof is based on critical point theory and a certain decomposition of the Sobolev space \(W^{1,2}(\Omega )\).

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

References

  1. Auchmuty, G., Steklov eigenproblems and the representation of solutions of elliptic boundary value problems. Numer. Funct. Anal. Optim., 2005, 25(3-4), 321–348.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartsch, T., Willem, M., On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc., 1995, 123(11), 3555–3561.

    Article  MathSciNet  MATH  Google Scholar 

  3. Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L., Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl., 1995, 2(4), 553–572.

    Article  MathSciNet  MATH  Google Scholar 

  4. Calderon, A.P., On an inverse boundary value problem. Comput. Appl. Math., 2006, 25(2–3), 133–138.

    MathSciNet  MATH  Google Scholar 

  5. Chabrowski, J., Tintarev, C., An elliptic problem with an indefinite nonlinearity and a parameter in the boundary condition. NoDEA Nonlinear Differ. Equ. Appl., 2014, 21(4), 519–540.

    Article  MathSciNet  MATH  Google Scholar 

  6. Cherrier, P., Problèmes de Neumann non linéaires sur les variétés Riemanniennes. J. Funct. Anal., 1984, 57, 154–206

    Article  MathSciNet  MATH  Google Scholar 

  7. Chipot, M., Shafrir, I., Fila, M., On the solutions to some elliptic equations with nonlinear boundary conditions. Adv. Differ. Equ., 1996, 1(1), 91–110.

    MathSciNet  MATH  Google Scholar 

  8. Escobar, J.F., The geometry of the first non-zero Steklov eigenvalue. J. Funct. Anal., 1997, 150, 544–556.

    Article  MathSciNet  MATH  Google Scholar 

  9. Escobar, J.F., A comparison theorem for the first non-zero Steklov eigenvalue. J. Funct. Anal., 2000, 178, 143–155.

    Article  MathSciNet  MATH  Google Scholar 

  10. Flores, C., del Pino, M., Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains. Commun. Partial. Differ. Equ., 2001, 26(11–12), 2189–2210.

    MathSciNet  MATH  Google Scholar 

  11. Garcia-Azorero, J., Peral, I., Rossi, J.D., A convex–concave problem with a nonlinear boundary condition. J. Differ. Equ., 2004, 198, 91–128.

    Article  MathSciNet  MATH  Google Scholar 

  12. de Godoi, J.D., Miyagaki, O.H., Rodrigues, R.S., On Steklov–Neumann boundary value problems for some quasilinear elliptic equations. Appl. Math. Lett., 2015, 45, 47–51.

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu, B., Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition. Differ. Integral Equ., 1994, 7(2), 301–313.

    MathSciNet  MATH  Google Scholar 

  14. Kajikiya, R., Naimen, D., Two sequences of solutions for indefinite superlinear–sublinear elliptic equations with nonlinear boundary conditions. Comm. Pure. Appl. Anal., 2014, 13(4), 1593–1612.

    Article  MathSciNet  MATH  Google Scholar 

  15. Leadi, L., Marcos, A., A weighted eigencurve for Steklov problems with a potential. NoDEA Nonlinear Differ. Equ. Appl., 2013, 20(3), 687–713.

    Article  MathSciNet  MATH  Google Scholar 

  16. Mavinga, N., Nkashama, M.N., Steklov–Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions. J. Differ. Equ., 2010, 248(5), 1212–1229.

    Article  MathSciNet  MATH  Google Scholar 

  17. Mavinga, N., Generalized eigenproblem and nonlinear elliptic equations with nonlinear boundary conditions. Proc. R. Soc. Edinb. Sect. A, 2012, 142(1), 137–153.

    Article  MathSciNet  MATH  Google Scholar 

  18. Nittka, R.: Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differ. Equ. 251, 860–880 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pagani, C.D., Pierotti, D., Variational methods for nonlinear Steklov eigenvalue problems with an indefinite weight function. Calc. Var. Partial Differ. Equ., 2010, 39(1–2), 35–58.

    Article  MathSciNet  MATH  Google Scholar 

  20. Da Silva, E.D., De Paiva, F.O., Landesman-Lazer type conditions and multiplicity results for nonlinear elliptic problems with Neumann boundary values. Acta Math. Sin. (Engl. Ser.), 2014, 30(2), 229–250.

  21. Stancut, I.L., On the existence of infinitely many solutions of a nonlinear Neumann problem involving the m-Laplace operator. Ann. Univ. Craiova Math. Comput. Sci. Ser., 2015, 42(2), 402–426.

  22. Steklov, M.W., Sur les problmes fondamentaux de la physique mathmatique. Ann. Sci. Ec. Norm. Super., 1902, 19, 455–490.

    Article  Google Scholar 

  23. Terraccini, S., Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions. Differ. Integral Equ., 1995, 8(8), 1911–1922.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, NIT Rourkela, Rourkela, 769008, India

    Arun Kumar Badajena & Shesadev Pradhan

Authors
  1. Arun Kumar Badajena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shesadev Pradhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arun Kumar Badajena.

Additional information

Communicated by G.D. Veerappa Gowda.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Badajena, A.K., Pradhan, S. Existence of infinitely many solutions of nonlinear Steklov–Neumann problem. Indian J Pure Appl Math 54, 447–455 (2023). https://doi.org/10.1007/s13226-022-00266-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-022-00266-1

Keywords

Search

Navigation