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January  2020, 19(1): 511-525. doi: 10.3934/cpaa.2020025

## Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator

 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Received  January 2019 Revised  May 2019 Published  July 2019

Fund Project: This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.307.

We consider the boundary value problem
 $\begin{equation*} \begin{cases} -{\rm div}_G(w_1\nabla_G u) = w_2f(u) &\text{ in } \Omega,\\ u=0 &\text{ on } \partial\Omega, \end{cases}\end{equation*}$
where
 $\Omega$
is a bounded or unbounded
 $C^1$
domain of
 $\mathbb{R}^N$
,
 $w_1, w_2 \in L^1_{\rm loc}(\Omega)\setminus\{0\}$
are nonnegative functions,
 $f$
is an increasing function,
 $\nabla_G$
and
 ${\rm div}_G$
are Grushin gradient and Grushin divergence, respectively. We prove some Liouville theorems for stable weak solutions of the problem under suitable assumptions on
 $\Omega$
,
 $w_1$
,
 $w_2$
and
 $f$
. We also show the sharpness of our results when
 $\Omega=\mathbb{R}^N$
and
 $f$
has power or exponential growth.
Citation: Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure and Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025
##### References:
 [1] C. T. Anh, J. Lee and B. K. My, On the classification of solutions to an elliptic equation involving the {G}rushin operator, Complex Var. Elliptic Equ., 63 (2018), 671-688.  doi: 10.1080/17476933.2017.1332051. [2] I. Birindelli, I. Capuzzo Dolcetta and A. Cutrì, Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 295-308.  doi: 10.1016/S0294-1449(97)80138-2. [3] I. Birindelli and J. Prajapat, Nonlinear Liouville theorems in the Heisenberg group via the moving plane method, Comm. Partial Differential Equations, 24 (1999), 1875-1890.  doi: 10.1080/03605309908821485. [4] D. Castorina, P. Esposito and B. Sciunzi, Low dimensional instability for semilinear and quasilinear problems in $\Bbb R^N$, Commun. Pure Appl. Anal., 8 (2009), 1779-1793.  doi: 10.3934/cpaa.2009.8.1779. [5] C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.  doi: 10.1090/S0002-9939-2011-11351-0. [6] L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1099-1119.  doi: 10.1016/j.anihpc.2008.06.001. [7] L. D'Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc., 132 (2004), 725-734.  doi: 10.1090/S0002-9939-03-07232-0. [8] E. N. Dancer, Y. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005. [9] A. T. Duong and N. T. Nguyen, Liouville type theorems for elliptic equations involving Grushin operator and advection, Electron. J. Differential Equations, Paper No. 108, 11. [10] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u = f(u)$ in $\Bbb R^N$, J. Eur. Math. Soc. (JEMS), 12 (2010), 855-882.  doi: 10.4171/JEMS/217. [11] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, vol. 143 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802. [12] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\Bbb R^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001. [13] A. Farina, Stable solutions of $-\Delta u = e^u$ on $\Bbb R^N$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021. [14] B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations, 19 (1994), 523-604.  doi: 10.1080/03605309408821025. [15] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508. [16] P. Le, Liouville theorems for stable solutions of p-Laplace equations with convex nonlinearities, J. Math. Anal. Appl., 443 (2016), 431-444.  doi: 10.1016/j.jmaa.2016.05.040. [17] P. Le and V. Ho, Stable solutions to weighted quasilinear problems of Lane-Emden type, Electron. J. Differential Equations, Paper No. 71, 11. [18] P. Le and V. Ho, Liouville results for stable solutions of quasilinear equations with weights, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 357-368. [19] P. Le, D. H. T. Le and K. A. T. Le, On stable solutions to weighted quasilinear problems of Gelfand type, Mediterr. J. Math., 15 (2018), Art. 94, 12. doi: 10.1007/s00009-018-1143-7. [20] P. Le, H. T. Nguyen and T. Y. Nguyen, On positive stable solutions to weighted quasilinear problems with negative exponent, Complex Var. Elliptic Equ., 63 (2018), 1739-1751.  doi: 10.1080/17476933.2017.1403429. [21] D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc. (JEMS), 12 (2010), 611-654.  doi: 10.4171/JEMS/210. [22] B. Rahal, Liouville-type theorems with finite {M}orse index for semilinear $\Delta_\lambda$-Laplace operators, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 21, 19. doi: 10.1007/s00030-018-0512-z. [23] C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017. [24] X. Yu, Liouville type theorem for nonlinear elliptic equation involving Grushin operators, Commun. Contemp. Math., 17 (2015), 1450050, 12. doi: 10.1142/S0219199714500503.

show all references

##### References:
 [1] C. T. Anh, J. Lee and B. K. My, On the classification of solutions to an elliptic equation involving the {G}rushin operator, Complex Var. Elliptic Equ., 63 (2018), 671-688.  doi: 10.1080/17476933.2017.1332051. [2] I. Birindelli, I. Capuzzo Dolcetta and A. Cutrì, Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 295-308.  doi: 10.1016/S0294-1449(97)80138-2. [3] I. Birindelli and J. Prajapat, Nonlinear Liouville theorems in the Heisenberg group via the moving plane method, Comm. Partial Differential Equations, 24 (1999), 1875-1890.  doi: 10.1080/03605309908821485. [4] D. Castorina, P. Esposito and B. Sciunzi, Low dimensional instability for semilinear and quasilinear problems in $\Bbb R^N$, Commun. Pure Appl. Anal., 8 (2009), 1779-1793.  doi: 10.3934/cpaa.2009.8.1779. [5] C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.  doi: 10.1090/S0002-9939-2011-11351-0. [6] L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1099-1119.  doi: 10.1016/j.anihpc.2008.06.001. [7] L. D'Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc., 132 (2004), 725-734.  doi: 10.1090/S0002-9939-03-07232-0. [8] E. N. Dancer, Y. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005. [9] A. T. Duong and N. T. Nguyen, Liouville type theorems for elliptic equations involving Grushin operator and advection, Electron. J. Differential Equations, Paper No. 108, 11. [10] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u = f(u)$ in $\Bbb R^N$, J. Eur. Math. Soc. (JEMS), 12 (2010), 855-882.  doi: 10.4171/JEMS/217. [11] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, vol. 143 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802. [12] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\Bbb R^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001. [13] A. Farina, Stable solutions of $-\Delta u = e^u$ on $\Bbb R^N$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021. [14] B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations, 19 (1994), 523-604.  doi: 10.1080/03605309408821025. [15] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508. [16] P. Le, Liouville theorems for stable solutions of p-Laplace equations with convex nonlinearities, J. Math. Anal. Appl., 443 (2016), 431-444.  doi: 10.1016/j.jmaa.2016.05.040. [17] P. Le and V. Ho, Stable solutions to weighted quasilinear problems of Lane-Emden type, Electron. J. Differential Equations, Paper No. 71, 11. [18] P. Le and V. Ho, Liouville results for stable solutions of quasilinear equations with weights, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 357-368. [19] P. Le, D. H. T. Le and K. A. T. Le, On stable solutions to weighted quasilinear problems of Gelfand type, Mediterr. J. Math., 15 (2018), Art. 94, 12. doi: 10.1007/s00009-018-1143-7. [20] P. Le, H. T. Nguyen and T. Y. Nguyen, On positive stable solutions to weighted quasilinear problems with negative exponent, Complex Var. Elliptic Equ., 63 (2018), 1739-1751.  doi: 10.1080/17476933.2017.1403429. [21] D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc. (JEMS), 12 (2010), 611-654.  doi: 10.4171/JEMS/210. [22] B. Rahal, Liouville-type theorems with finite {M}orse index for semilinear $\Delta_\lambda$-Laplace operators, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 21, 19. doi: 10.1007/s00030-018-0512-z. [23] C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017. [24] X. Yu, Liouville type theorem for nonlinear elliptic equation involving Grushin operators, Commun. Contemp. Math., 17 (2015), 1450050, 12. doi: 10.1142/S0219199714500503.
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