Skip to main content
SpringerLink
Log in

Existence and Multiplicity Results for Anisotropic Double-Phase Differential Inclusion with Unbalanced Growth and Lack of Compactness

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider a class of anisotropic double-phase differential inclusion problems driven by \((p_1(x);p_2(x))\)-Laplacian like operators:

$$\begin{aligned}{} & {} -\text {div}\left[ \phi (x,|\nabla u|)\nabla u+\psi (x,|\nabla u|)\nabla u\right] \\{} & {} \quad +\gamma (x)\left[ \phi (x,|u|)u+\psi (x,|u|)u\right] \in \eta (x)\partial R(x,u) \text{ in } \mathbb {R}^N. \end{aligned}$$

The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p(x)-Laplace operator, the generalized mean curvature operator or the capillarity differential operator with variable exponent. For these problems, we reveal the existence and multiplicity of two different groups of solutions (low and high energy solutions). The proofs rely on variational principle for locally Lipschitz functions combined with the properties of generalized Lebesgue–Sobolev spaces, on the nonsmooth symmetric Mountain-Pass Lemma, the Fountain Theorem and energy estimates.

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

Data Availability Statement

The authors declare that data supporting the findings of this study are available within the article and its supplementary information files.

References

  1. Bahrouni, A., Rădulescu, V., Repovš, D.: A weighted anisotropic variant of the Caffarelli–Kohn–Nirenberg inequality and applications. Nonlinearity 31, 1518–1534 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baraket, S., Chebbi, S., Chorfi, N., Rădulescu, V.: Non-autonomous eigenvalue problems with variable \((p_1, p_2)\)-growth. Adv. Nonlinear Stud. 17(4), 781–792 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)

    Google Scholar 

  4. Chang, K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80(1), 102–129 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dai, G.: Nonsmooth version of Fountain theorem and its application to a Dirichlet-type differential inclusion problem. Nonlinear Anal. 72(3–4), 1454–1461 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diening, L., Hästö, P., Harjulehto, P., Ružička, M.: Lebesgue and Sobolev Spaces with Variable Exponents, Springer Lecture Notes, vol. 2017. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  8. Edmunds, D.E., Rákosník, J.: Sobolev embeddings with variable exponent. Stud. Math. 143(3), 267–293 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan, X., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces \(W^{k, p(x)}(\Omega )\). J. Math. Anal. Appl. 262(2), 749–760 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, X., Zhang, Q.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gasiński, L., Papageorgiou, N. S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Ser. Math. Anal. Appl., vol. 8. Chapman & Hall/CRC, Boca Raton (2005)

  12. Ge, B., Rădulescu, V.: Infinitely many solutions for a non-homogeneous differential inclusion with lack of compactness. Adv. Nonlinear Stud. 19(3), 625–637 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jabri, Y.: The Mountain Pass Theorem Variants, Generalizations and Some Applications, Encyclopedia of Mathematics and Its applications. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  14. Kim, I.H., Kim, Y.H.: Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents. Manuscr. Math. 147, 169–191 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537–554 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Mingione, G., Rădulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. 501, 125197 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  18. Papageorgiou, N. S., Rădulescu, V., Repovs̆, D. D.: Nonlinear Analysis—Theory and Methods, Springer Monogr. Math. Springer, Cham (2019)

  19. Pucci, P., Rădulescu, V.: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. 3(9), 543–582 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Pucci, P., Rădulescu, V.: The maximum principle with lack of monotonicity. Electron. J. Qual. Theory Differ. Equ. (58) , 1–11 (2018)

  21. Rădulescu, V.: Mountain pass theorems for nondifferentiable functions and applications. Proc. Japan Acad. Ser. A Math. Sci. 69(6), 193–198 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rădulescu, V.: A Lusternik–Schnirelmann type theorem for locally Lipschitz functionals with applications to multivalued periodic problems. Proc. Japan Acad. Ser. A Math. Sci. 71(7), 164–167 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Rădulescu, V., Repovs̆, D.: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis. CRC Press, Taylor & Francis Group, Boca Raton (2015)

  25. Rădulescu, V., Stăncuţ, I.L.: Combined concave-convex effects in anisotropic elliptic equations with variable exponent. Nonlinear Differ. Equ. Appl. 22, 391–410 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Repovs̆, D.: Stationary waves of Schrödinger-type equations with variable exponent. Anal. Appl. (Singap.) 13(6), 645–661 (2015)

  27. Repovs̆, D.: Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete Contin. Dyn Syst. Ser. S 12(2), 401–411 (2019)

  28. Rodrigues, M.: Multiplicity of solutions on a nonlinear eigenvalue problem for \(p(x)\)-Laplacian-like operators. Mediterr. J. Math. 9, 211–223 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Uţă, V.F.: Multiple solutions for eigenvalue problems involving an indefinite potential and with (p1(x); p2(x)) balanced growth. Analele Ştiinţifice ale Universităţii “Ovidius” Constanţa Seria Matematica 27(1), 289–307 (2019)

  30. Uţă, V.F.: On the existence and multiplicity of eigenvalues for a class of double-phase non-autonomous problems with variable exponent growth. Electron. J. Qual. Theory Differ. Equ. 28, 1–22 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Uţă, V.F.: Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term. Electron. J. Qual. Theory Differ. Equ. 5, 1–22 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, K.Q., Zhou, M.: Infinitely many weak solutions for \(p(x)\)-Laplacian-like problems with sign-changing potential. Electron. J. Qual. Theory Differ. Equ. 10, 1–14 (2020)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of V. F. Uta have been supported by a grant of the Romanian Ministry of Research, Innovation and Digitalization (MCID), project number 22 - Nonlinear Differential Systems in Applied Sciences, within PNRR-III-C9-2022-I8.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Vasile Florin Uţă

Authors
  1. Vasile Florin Uţă
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The contents and the results described by the paper were written and developed by its only author.

Corresponding author

Correspondence to Vasile Florin Uţă.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Uţă, V.F. Existence and Multiplicity Results for Anisotropic Double-Phase Differential Inclusion with Unbalanced Growth and Lack of Compactness. Mediterr. J. Math. 20, 267 (2023). https://doi.org/10.1007/s00009-023-02470-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02470-7

Keywords

Mathematics Subject Classification

Search

Navigation