Skip to main content
SpringerLink
Log in

Existence of Infinitely Many Solutions for Δγ-Laplace Problems

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In this article, we study the existence of infinitelymany solutions for the boundary–value problem

$$ - {\Delta _\gamma }u + a\left( x \right)u = f\left( {x,u} \right)in\Omega ,u = 0on\partial \Omega $$

, where Ω is a bounded domain with smooth boundary in ℝN (N ≥ 2) and Δγ is a subelliptic operator of the form

$${\Delta _\gamma }: = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}} \right)} ,{\partial _{{x_j}}}: = \frac{\partial }{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \cdots ,\gamma N} \right)$$

. Our main tools are the local linking and the symmetric mountain pass theorem in critical point theory.

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. A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” J. Funct. Anal. 14, 349–381 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems in Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2007), Vol.104.

  3. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differ. Equations in CBMS Regional Conference Series in Mathematics (Amer.Math. Soc., Providence, RI, 1986), Vol.65.

  4. M. Schechter and W. Zou, “Superlinear problems,” Pacific J.Math. 214 (1), 145–160 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  5. O. H. Miyagaki and M. A. S. Souto, “Superlinear problems without Ambrosetti and Rabinowitz growth condition,” J. Differ. Equations 245 (12), 3628–3638 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Lam and G. Lu, “N-Laplace equations in RN with subcritical and critical growth without the Ambrosetti–Rabinowitz condition,” Adv. Nonlinear Stud. 13 (2), 289–308 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Lam and G. Lu, “Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition,” J. Geom. Anal. 24 (1), 118–143 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Liu, “On superlinear problems without the Ambrosetti and Rabinowitz condition,” Nonlinear Anal. 73 (3), 788–795 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  9. Z. Liu and Z. Q. Wang, “On the Ambrosetti–Rabinowitz superlinear condition,” 4 (4), 563–574 (2004).

    Google Scholar 

  10. V. V. Grushin, “A certain class of hypoelliptic operators,” Mat. Sb. (N. S.) 83 (125), 456–473 (1970).

    MathSciNet  Google Scholar 

  11. N. M. Tri, Semilinear Degenerate Elliptic Differ. Equations, Local and Global Theories (Lambert Academic Publishing, 2010).

    Google Scholar 

  12. N. M. Tri, Recent Progress in the Theory of Semilinear Equations Involving Degenerate Elliptic Differential Operators (Publishing House for Science and Technology of the Vietnam Academy of Science and Technology, 2014).

    Google Scholar 

  13. C. T. Anh and B. K. My, “Existence of solutions to Δλ-Laplace equations without the Ambrosetti–Rabinowitz condition,” Complex Var. Elliptic Equ. 61 (1), 137–150 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  14. C. T. Anh and B. K. My, “Liouville–type theorems for elliptic inequalities involving the Δλ-Laplace operator,” Complex Var. Elliptic Equ. 61 (7), 1002–1013 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. E. Kogoj and E. Lanconelli, “On semilinear Δλ-Laplace equation,” Nonlinear Analysis. 75 (12), 4637–4649 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  16. D. T. Luyen, “Two nontrivial solutions of boundary value problems for semilinear Δγ differential equations,” Math. Notes 101 (5), 815–823 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  17. D. T. Luyen and N. M. Tri, “Existence of solutions to boundary value problems for semilinear Δγ differential equations,” Math. Notes 97 (1), 73–84 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  18. D. T. Luyen and N. M. Tri, “Large–time behavior of solutions to damped hyperbolic equation involving strongly degenerate elliptic differential operators,” Siberian Math. J. 57 (4), 632–649 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  19. D. T. Luyen and N. M. Tri, “Global attractor of the Cauchy problem for a semilinear degenerate damped hyperbolic equation involving the Grushin operator,” Ann. Pol.Math. 117 (2), 141–162 (2016).

    MathSciNet  MATH  Google Scholar 

  20. D. T. Luyen and N. M. Tri, “On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators” (in press).

  21. D. T. Luyen and N. M. Tri, “Existence of infinitely many solutions for semilinear degenerate Schrödinger equations,” J. Math. Anal. Appl. 461 (2), 1271–1286 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  22. P. T. Thuy and N. M. Tri, “Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations,” NoDEA Nonlinear Differ. Equations Appl. 19 (3), 279–298 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  23. P. T. Thuy and N. M. Tri, “Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators,” NoDEA Nonlinear Differ. Equations Appl. 20 (3), 1213–1224 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  24. N. M. Tri, “Critical Sobolev exponent for hypoelliptic operators,” ActaMath. Vietnam. 23 (1), 83–94 (1998).

    MATH  Google Scholar 

  25. G. Cerami, “An existence criterion for the critical points on unbounded manifolds,” Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (2), 332–336 (1978).

    MathSciNet  MATH  Google Scholar 

  26. S. Luan and A. Mao,“Periodic solutions for a class of non–autonomous Hamiltonian systems,” Nonlinear Anal. 61 (8), 1413–1426 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  27. S. J. Li and M. Willem, “Applications of local linking to critical point theory,” J. Math. Anal. Appl. 189 (1), 6–32 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  28. X. Wu, “Multiple solutions for quasilinear Schrödinger equations with a parameter,” J. Differ. Equations 256 (7), 2619–2632 (2014).

    Article  MATH  Google Scholar 

  29. Y. Wu and T. An,“Infinitely many solutions for a class of semilinear elliptic equations,” J. Math. Anal. Appl. 414 (1), 285–295 (2014).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hoa Lu University, Ninh Nhat, Ninh Binh City, Vietnam

    D. T. Luyen, D. T. Huong & L. T. H. Hanh

Authors
  1. D. T. Luyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. T. Huong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. T. H. Hanh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. T. Luyen.

Additional information

The article was submitted by the authors for the English version of the journal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luyen, D.T., Huong, D.T. & Hanh, L.T.H. Existence of Infinitely Many Solutions for Δγ-Laplace Problems. Math Notes 103, 724–736 (2018). https://doi.org/10.1134/S000143461805005X

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S000143461805005X

Keywords

Search

Navigation