• Previous Article
    The weak maximum principle for second-order elliptic and parabolic conormal derivative problems
  • CPAA Home
  • This Issue
  • Next Article
    Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian
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 & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025
References:
[1]

C. T. AnhJ. 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. Google Scholar

[2]

I. BirindelliI. 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. Google Scholar

[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. Google Scholar

[4]

D. CastorinaP. 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. Google Scholar

[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. Google Scholar

[6]

L. DamascelliA. FarinaB. 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. Google Scholar

[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. Google Scholar

[8]

E. N. DancerY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

B. FranchiC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

P. Le and V. Ho, Stable solutions to weighted quasilinear problems of Lane-Emden type, Electron. J. Differential Equations, Paper No. 71, 11. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

P. LeH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

X. Yu, Liouville type theorem for nonlinear elliptic equation involving Grushin operators, Commun. Contemp. Math., 17 (2015), 1450050, 12. doi: 10.1142/S0219199714500503. Google Scholar

show all references

References:
[1]

C. T. AnhJ. 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. Google Scholar

[2]

I. BirindelliI. 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. Google Scholar

[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. Google Scholar

[4]

D. CastorinaP. 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. Google Scholar

[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. Google Scholar

[6]

L. DamascelliA. FarinaB. 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. Google Scholar

[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. Google Scholar

[8]

E. N. DancerY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

B. FranchiC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

P. Le and V. Ho, Stable solutions to weighted quasilinear problems of Lane-Emden type, Electron. J. Differential Equations, Paper No. 71, 11. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

P. LeH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

X. Yu, Liouville type theorem for nonlinear elliptic equation involving Grushin operators, Commun. Contemp. Math., 17 (2015), 1450050, 12. doi: 10.1142/S0219199714500503. Google Scholar

[1]

SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026

[2]

Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011

[3]

M. Á. Burgos-Pérez, J. García-Melián, A. Quaas. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4703-4721. doi: 10.3934/dcds.2016004

[4]

Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167

[5]

Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377

[6]

Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489

[7]

Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206

[8]

Linfen Cao, Wenxiong Chen. Liouville type theorems for poly-harmonic Navier problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3937-3955. doi: 10.3934/dcds.2013.33.3937

[9]

Y. Kabeya, Eiji Yanagida, Shoji Yotsutani. Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems. Communications on Pure & Applied Analysis, 2002, 1 (1) : 85-102. doi: 10.3934/cpaa.2002.1.85

[10]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869

[11]

Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks & Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967

[12]

Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293

[13]

Yayun Li, Yutian Lei. On existence and nonexistence of positive solutions of an elliptic system with coupled terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1749-1764. doi: 10.3934/cpaa.2018083

[14]

Takahiro Hashimoto. Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients. Conference Publications, 2009, 2009 (Special) : 349-358. doi: 10.3934/proc.2009.2009.349

[15]

Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393

[16]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665

[17]

Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057

[18]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399

[19]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121

[20]

Takahiro Hashimoto. Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains. Conference Publications, 2005, 2005 (Special) : 376-385. doi: 10.3934/proc.2005.2005.376

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (31)
  • HTML views (84)
  • Cited by (0)

Other articles
by authors

[Back to Top]