## Abstract

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

where Ω is a bounded domain with smooth boundary in R^{N} (*N* ≥ 2) and Δ_{
γ
} is the subelliptic operator of the type

We use the three critical point theorem.

### Similar content being viewed by others

## References

D. Jerison, “The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II,” J. Funct. Anal.

**43**(2), 224–257 (1981).D. S. Jerison and J. M. Lee, “The Yamabe problem on CR manifolds,” J. Differential Geom.

**25**(2), 167–197 (1987).N. Garofalo and E. Lanconelli, “Existence and nonexistence results for semilinear equations on the Heisenberg group,” Indiana Univ. Math. J.

**41**(1), 71–98 (1992).N. M. Tri, “On the Grushin equation,” Mat. Zametki

**63**(1), 95–105 [Math. Notes**63**(1–2), 84–93 (1998)].N. M. Tri, “Critical Sobolev exponent for hypoelliptic operators,” ActaMath. Vietnam.

**23**(1), 83–94 (1998).N. T. C. Thuy and N. M. Tri, “Some existence and nonexistence results for boundary-value problems for semilinear elliptic degenerate operators,” Russ. J. Math. Phys.

**9**(3), 365–370 (2002).P. T. Thuy and N. M. Tri, “Nontrivial solutions to boundary-value problems for semilinear strongly degenerate elliptic differential equations,” NoDEA Nonlinear Differential Equations Appl.

**19**(3), 279–298 (2012).A. E. Kogoj and E. Lanconelli, “On semilinear Δ

_{λ}−Laplace equation,” Nonlinear Analysis.**75**(12), 4637–4649 (2012).C. T. Anh and B. K. My, “Existence of solutions to Δ

_{λ}−Laplace equations without the Ambrosetti–Rabinowitz condition,” Complex Var. Elliptic Equ.**61**(1), 137–150 (2016).C. T. Anh and B. K. My, “Liouville–type theorems for elliptic inequalities involving the Δ

_{λ}−Laplace operator,” Complex Var. Elliptic Equ.**61**(7), 1002–1013 (2016).D. T. Luyen and N. M. Tri, “Existence of solutions to boundary-value problems for semilinear Δ

_{γ}-differential equations,” Math. Notes**97**(1–2), 73–84 (2015).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).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).N. M. Tri,

*Semilinear Degenerate Elliptic Differential Equations, Local and Global Theories*(Lambert Academic Publishing, 2010).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).A. Ambrosetti and G. Mancini, “Sharp nonuniqueness results for some nonlinear problems,” Nonlinear Anal.

**3**(5), 635–645 (1979).S. Ahmad, “Multiple nontrivial solutions of resonant and nonresonant asymptotically linear problems,” Proc. Amer. Math. Soc.

**96**(3), 405–409 (1987).N. Hirano, “Multiple nontrivial solutions of semilinear elliptic equations,” Proc. Amer. Math. Soc.

**103**(2), 468–472 (1988).S. Robinson, “Multiple solutions for semilinear elliptic boundary-value problems at resonance,” Electron. J. Differential Equations

**1**, 1–14 (1995).J. B. Su, “Semilinear elliptic boundary-value problems with double resonance between two consecutive eigenvalues,” Nonlinear Anal. 48 (6), Ser. A: TheoryMethods, 881–895 (2002).

V. V. Grushin, “A certain class of hypoelliptic operators,” Mat. Sb. (N. S.)

**83**(125), 456–473 (1970) [in Russian].J. Q. Liu, “The Morse index of a saddle point,” Systems Sci. Math. Sci.

**2**(1), 32–39 (1989).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

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

## Rights and permissions

## About this article

### Cite this article

Luyen, D.T. Two nontrivial solutions of boundary-value problems for semilinear Δ_{
γ
}-differential equations.
*Math Notes* **101**, 815–823 (2017). https://doi.org/10.1134/S0001434617050078

Received:

Published:

Issue Date:

DOI: https://doi.org/10.1134/S0001434617050078