Skip to main content
SpringerLink
Log in

Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz–Sobolev spaces without Ambrosetti–Rabinowitz condition

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

The aim of this work is to establish the existence and multiplicity of solutions for the following class of quasilinear problems

$$\begin{aligned} - \text{ div }\big (\varepsilon ^{2}\phi (\varepsilon |\nabla u|)\nabla u\big ) + V(x)\phi (\vert u\vert )u = f(u)\quad \text{ in } \quad {\mathbb {R}}^{N}, \end{aligned}$$

where \(\varepsilon\) is a positive parameter, \(N\ge 2\), V, f are continuous functions satisfying some technical conditions and \(\phi\) is a \(C^{1}\)-function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140, 285–300 (1997)

    Article  Google Scholar 

  2. Cingolani, S., Lazzo, M.: Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. Topol. Methods. Nonlinear Anal. 10, 1–13 (1997)

    Article  MathSciNet  Google Scholar 

  3. del Pino, M., Felmer, P.L.: Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4(2), 121–137 (1996)

    Article  Google Scholar 

  4. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986)

    Article  MathSciNet  Google Scholar 

  5. Oh, Y.G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131(2), 223–253 (1990)

    Article  MathSciNet  Google Scholar 

  6. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)

    Article  MathSciNet  Google Scholar 

  7. Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys. 153(2), 229–244 (1993)

    Article  MathSciNet  Google Scholar 

  8. Alves, C.O., Figueiredo, G.M.: Existence and multiplicity of positive solutions to a p-Laplacian equation in \({\mathbb{R}}^{N}\). Differ. Integral Equ. 19(2), 143–162 (2006)

    MATH  Google Scholar 

  9. Alves, C.O., da Silva, A.R.: Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz–Sobolev space. J. Math. Phys. 57, 111502 (2016). https://doi.org/10.1063/1.4966534

    Article  MathSciNet  MATH  Google Scholar 

  10. DiBenedetto, E.: \(C^{1, \gamma }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1985)

    Article  Google Scholar 

  11. Fukagai, N., Ito, M., Narukawa, K.: Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on \({\mathbb{R}}^{N}\). Funk. Ekvac. 49(2), 235–267 (2006)

    Article  Google Scholar 

  12. Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann. Mat. Pura Appl. 186(3), 539–564 (2007)

    Article  MathSciNet  Google Scholar 

  13. Ait-Mahiout, K., Alves, C.O.: Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz–Sobolev spaces. Complex Var. Elliptic Equ. 62, 767–785 (2017). https://doi.org/10.1080/17476933.2016.1243669

    Article  MathSciNet  MATH  Google Scholar 

  14. Ait-Mahiout, K., Alves, C.O.: Multiple solutions for a class of quasilinear problems in Orlicz–Sobolev spaces. Asymptot. Anal. 104, 49–66 (2017). https://doi.org/10.3233/ASY-171428

    Article  MathSciNet  MATH  Google Scholar 

  15. Alves, C.O., Figueiredo, G.M., Santos, J.A.: Strauss and Lions type results for a class of Orlicz–Sobolev spaces and applications. Topol. Methods Nonlinear Anal. 44(2), 435–456 (2014)

    Article  MathSciNet  Google Scholar 

  16. Alves, C.O., Carvalho, M.L.M., Gonçalves, J.V.A.: On existence of solution of variational multivalued elliptic equations with critical growth via the Ekeland principle. Commun. Contemp. Math. (2014). https://doi.org/10.1142/S0219199714500382.

    Article  MathSciNet  Google Scholar 

  17. Azzollini, A., d’Avenia, P., Pomponio, A.: Quasilinear elliptic equations in \({\mathbb{R}}^{N}\) via variational methods and Orlicz–Sobolev embeddings. Calc. Var. Partial Differ. Equ. 49, 197–213 (2014)

    Article  Google Scholar 

  18. Bonanno, G., Bisci, G.M., Rădulescu, V.: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces. Nonlinear Anal. 75, 4441–4456 (2012)

    Article  MathSciNet  Google Scholar 

  19. Bonanno, G., Bisci, G.M., Rădulescu, V.: Existence and multiplicity of solutions for a quasilinear nonhomogeneous problems: an Orlicz–Sobolev space setting. J. Math. Anal. Appl. 330, 416–432 (2007)

    Article  MathSciNet  Google Scholar 

  20. Fukagai, N., Ito, M., Narukawa, K.: Quasilinear elliptic equations with slowly growing principal part and critical Orlicz–Sobolev nonlinear term. Proc. R. Soc. Edinburgh Sect. A 139(1), 73–106 (2009)

    Article  MathSciNet  Google Scholar 

  21. Le, V.K., Motreanu, D., Motreanu, V.V.: On a non-smooth eigenvalue problem in Orlicz–Sobolev spaces. Appl. Anal. 89(2), 229–242 (2010)

    Article  MathSciNet  Google Scholar 

  22. Mihailescu, M., Rădulescu, V., Repovš, D.: On a non-homogeneous eigenvalue problem involving a potential: an Orlicz–Sobolev space setting. J. Math. Pures Appl. 93, 132–148 (2010)

    Article  MathSciNet  Google Scholar 

  23. Mihailescu, M., Rădulescu, V.: Nonhomogeneous Neumann problems in Orlicz–Sobolev spaces. C. R. Acad. Sci. Paris Ser. I(346), 401–406 (2008)

    Article  MathSciNet  Google Scholar 

  24. Mihailescu, M., Rădulescu, V.: Existence and multiplicity of solutions for a quasilinear non-homogeneous problems: an Orlicz–Sobolev space setting. J. Math. Anal. Appl. 330, 416–432 (2007)

    Article  MathSciNet  Google Scholar 

  25. Rădulescu, V.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121, 336–369 (2015)

    Article  MathSciNet  Google Scholar 

  26. Rădulescu, V., Repovš, D.: Partial Differential Equations with Variable Exponents Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015)

    Book  Google Scholar 

  27. Repovš, D.: Stationary waves of Schrödinger-type equations with variable exponent. Anal. Appl. 13, 645–661 (2015)

    Article  MathSciNet  Google Scholar 

  28. Santos, J.A.: Multiplicity of solutions for quasilinear equations involving critical Orlicz–Sobolev nonlinear terms. Electron. J. Differ. Equ. 2013(249), 1–13 (2013)

    MathSciNet  Google Scholar 

  29. Santos, J.A., Soares, S.H.M.: Radial solutions of quasilinear equations in Orlicz–Sobolev type spaces. J. Math. Anal. Appl. 428, 1035–1053 (2015)

    Article  MathSciNet  Google Scholar 

  30. Alves, C.O., da Silva, A.R.: Multiplicity and concentration behavior of solutions for a quasilinear problem involving N-functions via penalization method. Electron. J. Differ. Equ. 2016(158), 1–24 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Adams, A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic Press, London (2003)

    MATH  Google Scholar 

  32. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)

    Book  Google Scholar 

  33. Rao, M.N., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1985)

    Google Scholar 

  34. Chung, N.T., Toan, H.Q.: On a nonlinear and non-homogeneous problem without (A–R) type condition in Orlicz–Sobolev spaces. Appl. Comput. Math. 219, 7820–7829 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994)

    Article  MathSciNet  Google Scholar 

  36. Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem set on \({\mathbb{R}}^N\). Proc. R. Soc. Edinburgh 129, 787–809 (1999)

    Article  MathSciNet  Google Scholar 

  37. Liu, S.B.: On superlinear problems without Ambrosetti and Rabinowitz condition. Nonlinear Anal. 73, 788–795 (2010)

    Article  MathSciNet  Google Scholar 

  38. Li, G., Yang, C.: The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti–Rabinowitz conditions. Nonlinear Anal. 72, 4602–4613 (2010)

    Article  MathSciNet  Google Scholar 

  39. Miyagaki, O.H., Souto, M.A.S.: Super-linear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equ. 245, 3628–3638 (2008)

    Article  Google Scholar 

  40. Schechter, M., Zou, W.: Superlinear problems. Pacific J. Math. 214, 145–160 (2004)

    Article  MathSciNet  Google Scholar 

  41. Struwe, M., Tarantello, G.: On multivortex solutions in Chern–Simons gauge theory. Boll. Unione Mat. Ital. Sez. B 1(8), 109–121 (1998)

    MathSciNet  MATH  Google Scholar 

  42. Lusternik, L., Schnirelmann, L.: Méthodes Topologiques dans les Problèmes Variationnels. Hermann, Paris (1934)

    MATH  Google Scholar 

  43. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol. 74. Springer, New York (1989)

    Book  Google Scholar 

  44. Willem, M.: Minimax Theorems. Birkhauser, Basel (1996)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire “Théorie du point fixe et Applications”, École Normale Supérieure, BP 92, BP 92, 16006, Kouba, Algiers, Algeria

    Karima Ait-Mahiout

  2. Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, Campina Grande, PB, CEP:58429-900, Brazil

    Claudianor O. Alves

Authors
  1. Karima Ait-Mahiout
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudianor O. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudianor O. Alves.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Additional information

C. O. Alves was partially supported by CNPq/Brazil Proc. 304804/2017-7.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ait-Mahiout, K., Alves, C.O. Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz–Sobolev spaces without Ambrosetti–Rabinowitz condition. J Elliptic Parabol Equ 4, 389–416 (2018). https://doi.org/10.1007/s41808-018-0026-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41808-018-0026-1

Keywords

Mathematics Subject Classification

Search

Navigation