Sharp criteria of Liouville type for some nonlinear systems

Yutian  Lei; Congming  Li

doi:10.3934/dcds.2016.36.3277

Previous Article
Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy
DCDS Home
This Issue
Next Article
Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients

June 2016, 36(6): 3277-3315. doi: 10.3934/dcds.2016.36.3277

Sharp criteria of Liouville type for some nonlinear systems

Yutian Lei ^1, and Congming Li ^2,

1.	Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
2.	Department of Applied Mathematical, University of Colorado at Boulder, Boulder, CO 80309, United States

Received June 2015 Revised October 2015 Published December 2015

Abstract
Related Papers

In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type. These nonexistence results, known as Liouville theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type of integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.

Keywords: integral equations, $\gamma$-Laplace equations., Liouville type theorems, Lane-Emden equations, Critical exponents.

Mathematics Subject Classification: Primary: 35B09, 35J48; Secondary: 35B06, 35B40, 35J60, 35B45, 45G05, 45G1.

Citation: Yutian Lei, Congming Li. Sharp criteria of Liouville type for some nonlinear systems. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3277-3315. doi: 10.3934/dcds.2016.36.3277

References:

[1]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar
[2]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar
[3]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. Google Scholar
[4]	A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Lett., 4 (1997), 91-102. doi: 10.4310/MRL.1997.v4.n1.a9. Google Scholar
[5]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar
[6]	W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: 10.2307/2951844. Google Scholar
[7]	W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. Google Scholar
[8]	W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. Google Scholar
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. Google Scholar
[10]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. Google Scholar
[11]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. Google Scholar
[12]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar
[13]	M. Franca, Classification of positive solutions of p-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434. Google Scholar
[14]	F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, Wilmington, NC, USA, 327-335. Google Scholar
[15]	F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x. Google Scholar
[16]	B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. Google Scholar
[17]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. Google Scholar
[18]	M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X. Google Scholar
[19]	C. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p=0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237. doi: 10.1017/S0308210500022708. Google Scholar
[20]	F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2. Google Scholar
[21]	L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187. doi: 10.5802/aif.944. Google Scholar
[22]	C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. Google Scholar
[23]	N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(\|Du\|^{m-2} Du)+K(\|x\|)u^q=0$ in $R^n$, J. Math. Soc. Japan, 45 (1993), 719-742. doi: 10.2969/jmsj/04540719. Google Scholar
[24]	T. Kilpelaiinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. Google Scholar
[25]	D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9. Google Scholar
[26]	Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758. doi: 10.1016/j.jde.2011.10.009. Google Scholar
[27]	Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547. Google Scholar
[28]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. Google Scholar
[29]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. Google Scholar
[30]	C. Li, A degree theory approach for the shooting method, arXiv:1301.6232v1, 2013. Google Scholar
[31]	C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. Google Scholar
[32]	Y.-Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. Google Scholar
[33]	Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p=0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330. doi: 10.1016/0022-0396(92)90034-K. Google Scholar
[34]	Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$, Duke Math. J., 57 (1988), 895-924. doi: 10.1215/S0012-7094-88-05740-7. Google Scholar
[35]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. Google Scholar
[36]	C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comm. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052. Google Scholar
[37]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. Google Scholar
[38]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270. Google Scholar
[39]	C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. Google Scholar
[40]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^n$, Differential Integral Equations, 9 (1996), 465-479. Google Scholar
[41]	W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei., 77 (1986), 231-257. Google Scholar
[42]	L. Nirenberg, Topics in Nonlinear Functional Analysis, Notes by R. A. Artino, Courant Institute of Mathematical Sciences, New York University, New York, 1974. Google Scholar
[43]	M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76 (1988), 140-159. doi: 10.1016/0022-1236(88)90053-5. Google Scholar
[44]	L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700. doi: 10.1090/S0002-9939-1987-0894440-8. Google Scholar
[45]	N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859. Google Scholar
[46]	P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036. Google Scholar
[47]	P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhauser Verlag, Basel, 2007. Google Scholar
[48]	J. Serrin and H. Zou, Non-existence of positive solution of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. Google Scholar
[49]	J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380. Google Scholar
[50]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math.,189 (2002), 79-142. doi: 10.1007/BF02392645. Google Scholar
[51]	Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014. Google Scholar
[52]	S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882. doi: 10.1016/j.jfa.2012.09.012. Google Scholar
[53]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. Google Scholar
[54]	X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z. Google Scholar

show all references

References:

[1]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar
[2]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar
[3]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. Google Scholar
[4]	A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Lett., 4 (1997), 91-102. doi: 10.4310/MRL.1997.v4.n1.a9. Google Scholar
[5]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar
[6]	W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: 10.2307/2951844. Google Scholar
[7]	W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. Google Scholar
[8]	W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. Google Scholar
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. Google Scholar
[10]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. Google Scholar
[11]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. Google Scholar
[12]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar
[13]	M. Franca, Classification of positive solutions of p-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434. Google Scholar
[14]	F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, Wilmington, NC, USA, 327-335. Google Scholar
[15]	F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x. Google Scholar
[16]	B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. Google Scholar
[17]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. Google Scholar
[18]	M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X. Google Scholar
[19]	C. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p=0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237. doi: 10.1017/S0308210500022708. Google Scholar
[20]	F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2. Google Scholar
[21]	L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187. doi: 10.5802/aif.944. Google Scholar
[22]	C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. Google Scholar
[23]	N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(\|Du\|^{m-2} Du)+K(\|x\|)u^q=0$ in $R^n$, J. Math. Soc. Japan, 45 (1993), 719-742. doi: 10.2969/jmsj/04540719. Google Scholar
[24]	T. Kilpelaiinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. Google Scholar
[25]	D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9. Google Scholar
[26]	Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758. doi: 10.1016/j.jde.2011.10.009. Google Scholar
[27]	Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558. doi: 10.3934/dcds.2011.30.547. Google Scholar
[28]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. Google Scholar
[29]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. Google Scholar
[30]	C. Li, A degree theory approach for the shooting method, arXiv:1301.6232v1, 2013. Google Scholar
[31]	C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. Google Scholar
[32]	Y.-Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. Google Scholar
[33]	Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p=0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330. doi: 10.1016/0022-0396(92)90034-K. Google Scholar
[34]	Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$, Duke Math. J., 57 (1988), 895-924. doi: 10.1215/S0012-7094-88-05740-7. Google Scholar
[35]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. Google Scholar
[36]	C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comm. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052. Google Scholar
[37]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. Google Scholar
[38]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270. Google Scholar
[39]	C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. Google Scholar
[40]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^n$, Differential Integral Equations, 9 (1996), 465-479. Google Scholar
[41]	W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei., 77 (1986), 231-257. Google Scholar
[42]	L. Nirenberg, Topics in Nonlinear Functional Analysis, Notes by R. A. Artino, Courant Institute of Mathematical Sciences, New York University, New York, 1974. Google Scholar
[43]	M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76 (1988), 140-159. doi: 10.1016/0022-1236(88)90053-5. Google Scholar
[44]	L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700. doi: 10.1090/S0002-9939-1987-0894440-8. Google Scholar
[45]	N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859. Google Scholar
[46]	P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036. Google Scholar
[47]	P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhauser Verlag, Basel, 2007. Google Scholar
[48]	J. Serrin and H. Zou, Non-existence of positive solution of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. Google Scholar
[49]	J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380. Google Scholar
[50]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math.,189 (2002), 79-142. doi: 10.1007/BF02392645. Google Scholar
[51]	Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014. Google Scholar
[52]	S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882. doi: 10.1016/j.jfa.2012.09.012. Google Scholar
[53]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. Google Scholar
[54]	X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z. Google Scholar

[1]	Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094
[2]	Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011
[3]	Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510
[4]	Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017
[5]	Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058
[6]	Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167
[7]	Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469
[8]	Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021184
[9]	Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[10]	Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719
[11]	Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291
[12]	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
[13]	Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807
[14]	Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure & Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036
[15]	Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1681-1698. doi: 10.3934/cpaa.2021036
[16]	Dongho Chae, Shangkun Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5267-5285. doi: 10.3934/dcds.2016031
[17]	Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943
[18]	Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085
[19]	Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301
[20]	Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4185-4206. doi: 10.3934/dcds.2021033

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

Sharp criteria of Liouville type for some nonlinear systems

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Sharp criteria of Liouville type for some nonlinear systems

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors