Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations


In this paper, we study the existence of multiple solutions for the boundary value problem

$$\begin{aligned} \begin{array}{llll} -\Delta _{\gamma } u&{}= f(x,u) + g(x,u) &{} \text{ in } &{} \Omega , \\ u&{}= 0 &{} \text{ on } &{} \partial \Omega , \end{array} \end{aligned}$$

where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb {R}^N \ (N \ge 2),\) \(f(x,\xi ), g(x,\xi )\) are Carathéodory functions, \(f(x,\xi )\) is odd in \(\xi\), \(g(x,\xi )\) is perturbation term and \(\Delta _{\gamma }\) is the strongly degenerate elliptic operator of the type

$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma : = (\gamma _1, \gamma _2,\ldots , \gamma _N). \end{aligned}$$

We use the minimax method and Rabinowitz’s perturbation method. This result is a generalization of that of Luyen and Tri (Complex Var Elliptic Equ 64(6):1050–1066, 2019).

This is a preview of subscription content, access via your institution.


  1. 1.

    Bahri, A., Berestycki, H.: A perturbation method in critical point theory and applications. Trans. Am. Math. Soc. 267(1), 1–32 (1981)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Struwe, M.: Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscr. Math. 32(3–4), 335–364 (1980)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bahri, A., Lions, P.L.: Morse index of some min–max critical points. I. Application to multiplicity results. Commun. Pure Appl. Math. 41(8), 1027–1037 (1988)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Rabinowitz, P.H.: Multiple critical points of perturbed symmetric functionals. Trans. Am. Math. Soc. 272(2), 753–769 (1982)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986). viii+100 pp

  6. 6.

    Tehrani, H.T.: Infinitely many solutions for indefinite semilinear elliptic equations without symmetry. Commun. Partial Differ. Equ. 21(3–4), 541–557 (1996)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bolle, P., Ghoussoub, N., Tehrani, H.: The multiplicity of solutions in non-homogeneous boundary value problems. Manuscr. Math. 101(3), 325–350 (2000)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Long, Y.: Multiple solutions of perturbed superquadratic second order Hamiltonian systems. Trans. Am. Math. Soc. 311(2), 749–780 (1989)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Hirano, N., Zou, W.: A perturbation method for multiple sign-changing solutions. Calc. Var. Partial Differ. Equ. 37(1–2), 87–98 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Tanaka, K.: Morse indices at critical points related to the symmetric mountain pass theorem and applications. Commun. Partial Differ. Equ. 14(1), 99–128 (1989)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Jerison, D.: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II. J. Funct. Anal. 43(2), 224–257 (1981)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Jerison, D.S., Lee, J.M.: The Yamabe problem on CR manifolds. J. Differ. Geom. 25(2), 167–197 (1987)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Garofalo, N., Lanconelli, E.: Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Univ. Math. J. 41(1), 71–98 (1992)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Tri, N.M.: Critical Sobolev exponent for hypoelliptic operators. Acta Math. Vietnam 23(1), 83–94 (1998)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Franchi, B., Lanconelli, E.: An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality. Commun. Partial Differ. Equ. 9(13), 1237–1264 (1984)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Kogoj, A.E., Lanconelli, E.: On semilinear \(\Delta _\lambda -\)Laplace equation. Nonlinear Anal. 75(12), 4637–4649 (2012)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Grushin, V.V.: A certain class of hypoelliptic operators. Mat. Sb. (N. S.) 83(125), 456–473 (1970) (in Russian)

    MathSciNet  Google Scholar 

  18. 18.

    Tri, N.M.: On the Grushin equation. Mat. Zametki 63(1), 95–105 (1998)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Thuy, N.T.C., Tri, N.M.: Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators. Russ. J. Math. Phys. 9(3), 365–370 (2002)

    MathSciNet  MATH  Google Scholar 

  20. 20.

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

    MathSciNet  Article  Google Scholar 

  21. 21.

    Tri, N.M.: 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, Hanoi (2014)

    Google Scholar 

  22. 22.

    Tri, N.M.: Semilinear Degenerate Elliptic Differential Equations, Local and Global Theories, p. 271. Lambert Academic Publishing, Saarbrücken (2010)

    Google Scholar 

  23. 23.

    Anh, C.T., My, B.K.: Existence of solutions to \(\Delta _\lambda -\)Laplace equations without the Ambrosetti-Rabinowitz condition. Complex Var. Elliptic Equ. 61(1), 137–150 (2016)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Bahrouni, A., Rǎdulescu, V.D., Repovš, D.: Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves. Nonlinearity 32(7), 2481–2495 (2019)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Bahrouni, A., Rǎdulescu, V.D., Winkert, P.: Double phase problems with variable growth and convection for the Baouendi–Grushin operator. Z. Angew. Math. Phys. 71(6), 183 (2020)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Luyen, D.T.: Two nontrivial solutions of boundary value problems for semilinear \(\Delta _{\gamma }\) differential equations. Math. Notes 101(5), 815–823 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Luyen, D.T., Huong, D.T., Hong, L.T.H.: Existence of infinitely many solutions for \(\Delta _\gamma -\)Laplace problems. Math. Notes 103(5), 724–736 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Luyen, D.T., Tri, N.M.: Existence of solutions to boundary value problems for semilinear \(\Delta _{\gamma }\) differential equations. Math. Notes 97(1), 73–84 (2015)

    MathSciNet  Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 30.

    Luyen, D.T., Tri, N.M.: 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 

  31. 31.

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

    MathSciNet  Article  Google Scholar 

  32. 32.

    Zou, W., Li, X.: Existence results for nonlinear degenerate elliptic equations with lower order terms. Adv. Nonlinear Anal. 10(1), 301–310 (2021)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Luyen, D.T., Tri, N.M.: On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators. Complex Var. Elliptic Equ. 64(6), 1050–1066 (2019)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Luyen, D.T., Tri, N.M.: Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator. Complex Var. Elliptic Equ. 65(12), 2135–2150 (2020)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston Inc., Boston, MA (1996)

    Google Scholar 

  36. 36.

    Mugnai, D.: Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differ. Equ. Appl. 11(3), 379–391 (2004) (a comment on the generalized Ambrosetti–Rabinowitz condition. NoDEA Nonlinear Differ. Equ. Appl. 19(3), 29–301 (2012))

    Article  Google Scholar 

  37. 37.

    Struwe, M.: Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 34, xvi+272pp. Springer, Berlin (1996)

  38. 38.

    Chen, H., Chen, H.G., Li, J.N.: Estimates of Dirichlet eigenvalues for degenerate \(\Delta _\mu -\)Laplace operators. Calc. Var. Partial Differ. Equ. 59(4), 27 (2020)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Chen, H., Chen, H.G., Li, J.N.: Research announcements on “Estimates of Dirichlet eigenvalues for degenerate \(\Delta _\mu -\)Laplace operators’’. J. Math. (PRC) 40(2), 127–130 (2020)

    Google Scholar 

Download references


The author warmly thanks the anonymous referees for the careful reading of the manuscript and for their useful and nice comments. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02–2020.13.

Author information



Corresponding author

Correspondence to Duong Trong Luyen.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Luyen, D.T., Van Cuong, P. Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations. Rend. Circ. Mat. Palermo, II. Ser (2021).

Download citation


  • Semilinear strongly degenerate elliptic equations
  • Boundary value problems
  • Critical points
  • Perturbation methods
  • Multiple solutions

Mathematics Subject Classification

  • Primary 35J60
  • Secondary 35B33
  • 35J25
  • 35J70