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doi: 10.3934/cpaa.2021187
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## Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

 Mathematics Department, Faculty of Sciences and Arts, King Khalid University, Muhayil Asir, Saudi Arabia, University of Tunis El Manar, Faculty of Sciences of Tunis Department of Mathematics, Campus University 2092 Tunis, Tunisia

Received  March 2021 Revised  September 2021 Early access November 2021

We examine the following degenerate elliptic system:
 $-\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v>0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!>\!0.$
We prove that the system has no stable solution provided
 $p, \theta >0$
and
 $N_s: = N_1+(1+s)N_2< 2 + \alpha + \beta,$
where
 $\alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}.$
This result is an extension of some results in [15]. In particular, we establish a new integral estimate for
 $u$
and
 $v$
(see Proposition 1.1), which is crucial to deal with the case
 $0 < p < 1.$
Citation: Foued Mtiri. Liouville type theorems for stable solutions of elliptic system involving the Grushin operator. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021187
##### References:
 [1] I. Birindelli, I. Capuzzo Dolcetta and A. Cutr$\grave{\rm{l}}$, 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.  Google Scholar [2] I. Birindelli and J. Prajapat, Nonlinear Liouville theorems in the Heisenberg group via the moving plane method, Commun. Partial Differ. Equ., 26 (1999), 1875-1890.  doi: 10.1080/03605309908821485.  Google Scholar [3] J. Busca and R. Manasevich, A Liouville-type theorem for Lane-Emden systems, Indiana Uni. Math. J., 51 (2002), 37-52.  doi: 10.1512/iumj.2002.51.2160.  Google Scholar [4] C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2013), 2357-2371.  doi: 10.1088/0951-7715/26/8/2357.  Google Scholar [5] A. T. Duong and Q. H. Phan, Liouville type theorem for nonlinear elliptic system involving Grushin operator, J. Math. Anal. Appl., 454 (2017), 785-801.  doi: 10.1016/j.jmaa.2017.05.029.  Google Scholar [6] B. Franchi and E. Lanconelli., Une métrique associée à une classe d'opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino. 1983,105–114, conference on linear partial and pseudodifferential operators, Torino, 1982.  Google Scholar [7] H. Hajlaoui, A. Harrabi and F. Mtiri, Liouville theorems for stable solutions of the weighted Lane-Emden system, Discrete Contin. Dyn. Syst., 37 (2017), 265-279.  doi: 10.3934/dcds.2017011.  Google Scholar [8] L. Hu, Liouville type results for semi-stable solutions of the weighted Lane-Emden system, J. Math. Anal. Appl., 432 (2015), 429-440.  doi: 10.1016/j.jmaa.2015.06.032.  Google Scholar [9] A. E. Kogoj and E. Lanconelli, Liouville theorem for X-elliptic operators, Nonlinear Anal., 70 (2009), 2974-2985.  doi: 10.1016/j.na.2008.12.029.  Google Scholar [10] A. E. Kogoj and E. Lanconelli, On semilinear $\Delta_{\lambda}-$ Laplace equation, Nonlinear Anal., 75 (2012), 4637-4649.  doi: 10.1016/j.na.2011.10.007.  Google Scholar [11] A. E. Kogoj and E. Lanconelli, Linear and semilinear problems involving $\Delta_{\lambda}-$ Laplacians, J. Differ. Equ., 25 (2018), 167-178.   Google Scholar [12] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in RN, Differ. Integral Equ., 9 (1996), 465-479.   Google Scholar [13] E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar [14] D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc., 12 (2010), 611-654.  doi: 10.4171/JEMS/210.  Google Scholar [15] F. Mtiri and D. Ye, Liouville theorems for stable at infinity solutions of Lane-Emden system, Nonlinearity, 32 (2019), 910-926.  doi: 10.1088/1361-6544/aaf078.  Google Scholar [16] F. Mtiri, On the Classification of Solutions to a Weighted Elliptic System Involving the Grushin Operator, Acta Appl. Math., 174 (2021), 21 pp. doi: 10.1007/s10440-021-00425-2.  Google Scholar [17] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248.  Google Scholar [18] L. Negro and G. Metafune and C. Spina, $L^{P}$ estimates for Baouendi-Grushin operator, arXiv: 1907.10439v1. Google Scholar [19] P. Polá$\check{c}$ik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar [20] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380.   Google Scholar [21] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.   Google Scholar [22] M. A. S. Souto, A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems, Differ. Int. Equ, 8 (1995), 1245-1258.   Google Scholar [23] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [24] R. C. A. M. Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1992), 375-398.  doi: 10.1007/BF00375674.  Google Scholar [25] X. Yu, Liouville type theorem for nonlinear elliptic equation involving Grushin operators, Commun. Contemp. Math., 17 (2015), 12 pp. doi: 10.1142/S0219199714500503.  Google Scholar

show all references

##### References:
 [1] I. Birindelli, I. Capuzzo Dolcetta and A. Cutr$\grave{\rm{l}}$, 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.  Google Scholar [2] I. Birindelli and J. Prajapat, Nonlinear Liouville theorems in the Heisenberg group via the moving plane method, Commun. Partial Differ. Equ., 26 (1999), 1875-1890.  doi: 10.1080/03605309908821485.  Google Scholar [3] J. Busca and R. Manasevich, A Liouville-type theorem for Lane-Emden systems, Indiana Uni. Math. J., 51 (2002), 37-52.  doi: 10.1512/iumj.2002.51.2160.  Google Scholar [4] C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2013), 2357-2371.  doi: 10.1088/0951-7715/26/8/2357.  Google Scholar [5] A. T. Duong and Q. H. Phan, Liouville type theorem for nonlinear elliptic system involving Grushin operator, J. Math. Anal. Appl., 454 (2017), 785-801.  doi: 10.1016/j.jmaa.2017.05.029.  Google Scholar [6] B. Franchi and E. Lanconelli., Une métrique associée à une classe d'opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino. 1983,105–114, conference on linear partial and pseudodifferential operators, Torino, 1982.  Google Scholar [7] H. Hajlaoui, A. Harrabi and F. Mtiri, Liouville theorems for stable solutions of the weighted Lane-Emden system, Discrete Contin. Dyn. Syst., 37 (2017), 265-279.  doi: 10.3934/dcds.2017011.  Google Scholar [8] L. Hu, Liouville type results for semi-stable solutions of the weighted Lane-Emden system, J. Math. Anal. Appl., 432 (2015), 429-440.  doi: 10.1016/j.jmaa.2015.06.032.  Google Scholar [9] A. E. Kogoj and E. Lanconelli, Liouville theorem for X-elliptic operators, Nonlinear Anal., 70 (2009), 2974-2985.  doi: 10.1016/j.na.2008.12.029.  Google Scholar [10] A. E. Kogoj and E. Lanconelli, On semilinear $\Delta_{\lambda}-$ Laplace equation, Nonlinear Anal., 75 (2012), 4637-4649.  doi: 10.1016/j.na.2011.10.007.  Google Scholar [11] A. E. Kogoj and E. Lanconelli, Linear and semilinear problems involving $\Delta_{\lambda}-$ Laplacians, J. Differ. Equ., 25 (2018), 167-178.   Google Scholar [12] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in RN, Differ. Integral Equ., 9 (1996), 465-479.   Google Scholar [13] E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar [14] D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc., 12 (2010), 611-654.  doi: 10.4171/JEMS/210.  Google Scholar [15] F. Mtiri and D. Ye, Liouville theorems for stable at infinity solutions of Lane-Emden system, Nonlinearity, 32 (2019), 910-926.  doi: 10.1088/1361-6544/aaf078.  Google Scholar [16] F. Mtiri, On the Classification of Solutions to a Weighted Elliptic System Involving the Grushin Operator, Acta Appl. Math., 174 (2021), 21 pp. doi: 10.1007/s10440-021-00425-2.  Google Scholar [17] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248.  Google Scholar [18] L. Negro and G. Metafune and C. Spina, $L^{P}$ estimates for Baouendi-Grushin operator, arXiv: 1907.10439v1. Google Scholar [19] P. Polá$\check{c}$ik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar [20] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380.   Google Scholar [21] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.   Google Scholar [22] M. A. S. Souto, A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems, Differ. Int. Equ, 8 (1995), 1245-1258.   Google Scholar [23] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [24] R. C. A. M. Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1992), 375-398.  doi: 10.1007/BF00375674.  Google Scholar [25] X. Yu, Liouville type theorem for nonlinear elliptic equation involving Grushin operators, Commun. Contemp. Math., 17 (2015), 12 pp. doi: 10.1142/S0219199714500503.  Google Scholar
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