Skip to main content
SpringerLink
Log in

Ground State Sign-Changing Solution for a Large Class of \( p(x) \& q(x)\)-Quasilinear Elliptic Equations with Nonsmooth Nonlinearity

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

Abstract

In this paper, the following new class of problems is studied

where \(\Omega \) is a smooth bounded domain in \({\mathbb {R}}^N\) (\(N\ge 3\)), \(p\in C({\overline{\Omega }})\) such that \(2\le p(x)<N\) for all \(x\in {\overline{\Omega }}\), and a is \(C^1\) real function. Without assuming the \(C^1\) and the well-known Ambrosetti-Rabinowitz conditions on the reaction term f, we prove that problem \({\mathcal {P}}\) has a least-energy sign-changing weak solution. Note that, our approach is based on a variant of the quantitative deformation lemma and some new technical lemmas.

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

The authors declare that this paper has no associated data.

References

  1. Ambrosio, V., Figueiredo, G.M., Isernia, T., Bisci, G.M.: Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations. Adv. Nonlinear Stud. 19, 113–132 (2018)

    Article  MATH  Google Scholar 

  2. Ambrosio, V., Isernia, T.: Sign-changing solutions for a class of fractional Schrödinger equations with vanishing potentials. Lincei Mat. Appl. 29, 127–152 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Barile, S., Figueiredo, G.M.: Existence of least energy positive negative and nodal solutions for a class of p &q-problems with potentials vanishing at infinity. J. Math. Anal. Appls. 427, 1205–1233 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barile, S., Figueiredo, G.M.: Existence of a least energy nodal solution for a class of p &q-quasilinear elliptic equations. Adv. Nonlinear Stud. 14, 511–530 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartsch, T., Weth, T., Willem, M.: Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bartsch, T., Weth, T.: Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincaré Anal. Non-Linéaire 22, 259–281 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bartsch, T., Liu, Z., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Comm. Partial Differ. Equ. 29, 25–42 (2012)

    Article  MATH  Google Scholar 

  8. Bartsch, T., Weth, T.: A note on additional properties of sign changing solutions to superlinear elliptic equations. Topol. Methods Nonlinear Anal. 22, 1–14 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Batista, A.M., Furtado, M.F.: Positive and nodal solutions for a nonlinear Schrödinger-Poisson system with sign-changing potentials. Nonlinear Anal. 39, 142–156 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Berestycki, H., Lions, P.L.: Nonlinear scalar field equations, I Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–346 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bonanno, G., Chinni, A.: Discontinuous elliptic problems involving the \(p(x)\)-Laplacian. Mach. Nachr. 284, 639–652 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Castro, A., Cossio, J., Neuberger, J.: A sign-changing solution for a superlinear Dirichlet problem. Rocky Mountain J. Math. 27, 1041–1053 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chang, X., Nie, Z., Wang, Z.Q.: Sign-Changing Solutions of Fractional \(p\)-Laplacian Problems. Adv. Nonlinear Stud. 19, 29–53 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, S., Tang, X.: Ground state sign-changing solutions for elliptic equations with logarithmic nonlinearity. Acta Math. Hungar. 157, 27–38 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Corrêa, F.J.S.A., Carvalho, M.L.M., Goncalves, J.V.A., Silva, E.D.: Sign changing solutions for quasilinear superlinear elliptic problems. Quart. J. Math. 68, 391–420 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Volume 2017 of Lecture Notes in Mathematics. Springer, Berlin (2017)

  17. Edmunds, D.E., Ràkosnik, J.: Sobolev embedding with variable exponent Stud. Math. 143, 267–293 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Fan, X.L., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces \(W^{k, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Fan, X.L., Zhao, Y.Z., Zhang, Q.H.: A strong maximum principle for \(p(x)\)-Laplace equations. Chin. J. Contemp. Math. 24, 495–500 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Fan, X.L., Zhang, Q.H., Zhao, D.: Eigenvalues of p(x)-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302, 306–317 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Figueiredo, G.M., Santos, J.A.: Existence of least energy solution with two nodal domains for a generalized Kirchoff problem in an Orlicz-Sobolev space. Math. Nachr. 290, 583–603 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Figueiredo, G.M., Santos Júnior, J.R.: Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity. J. Math. Phys. 56, 051506 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Figueiredo, G.M., Nascimento, R.G.: Existence of a nodal solution with minimal energy for a Kirchhoff equation. Math. Nachr. 288, 48–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Harjulehto, P., Hästö, P.: Lebesgue points in variable exponent spaces. Ann. Acad. Sci. Fenn. Math. 29, 295–306 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Isernia, T.: Sign-changing solutions for a fractional Kirchhoff equation. Nonlinear Anal. 190, 111623 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kovàik, O., Ràkosnik, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41, 592–618 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Bisci, G.M., Rădulescu, V.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Part. Differ. Equ. 54, 2985–3008 (2015)

    Article  MATH  Google Scholar 

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

  30. Papageorgiou, N.S., Rădulescu, V.D., Repŏvs, D.D.: Ground state and nodal solutions for a class of double phase problems. Z. Angew. Math. Phys. 71, 1–15 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748, Springer-Verlag, Berlin, (2000)

  32. Willem, M.: Minimax Theorems Birkhauser, (1996)

  33. Xu, L., Chen, H.: Positive, negative and least energy nodal solutions for Kirchhoff equations in \({\mathbb{R} }^N\). Complex. Var. Elliptic. Equ. 66, 1676–1698 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR Izv. 29, 33–36 (1987)

    Article  MATH  Google Scholar 

Download references

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Mathematics Department, Faculty of Sciences, University of Monastir, 5019, Monastir, Tunisia

    Hlel Missaoui

Authors
  1. Hlel Missaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors reviewed the manuscript.

Corresponding author

Correspondence to Hlel Missaoui.

Ethics declarations

Conflict of Interest

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

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

Missaoui, H. Ground State Sign-Changing Solution for a Large Class of \( p(x) \& q(x)\)-Quasilinear Elliptic Equations with Nonsmooth Nonlinearity. Mediterr. J. Math. 20, 242 (2023). https://doi.org/10.1007/s00009-023-02447-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02447-6

Keywords

Mathematics Subject Classification

Search

Navigation