• Previous Article
    Some symmetry results for entire solutions of an elliptic system arising in phase separation
  • DCDS Home
  • This Issue
  • Next Article
    The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds
2014, 34(6): 2513-2533. doi: 10.3934/dcds.2014.34.2513

On the Hénon-Lane-Emden conjecture

1. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1, Canada

2. 

Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2

Received  September 2012 Revised  May 2013 Published  December 2013

We consider Liouville-type theorems for the following Hénon-Lane-Emden system \begin{eqnarray*} \left\{ \begin{array}{lcl} -\Delta u&=& |x|^{a}v^p \ \ in\ \ \mathbb{R}^n,\\ -\Delta v&=& |x|^{b}u^q \ \ in\ \ \mathbb{R}^n, \end{array}\right. \end{eqnarray*} when $p,q \ge 1,$ $pq\neq1$, $a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \frac{n+a}{p+1}+\frac{n+b}{q+1}>{n-2}$. We show that this is indeed the case in dimension $n=3$ provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case.
    Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $n\ge 3$ in the first case (resp., $n\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1< p < \frac{n+2+2a}{n-2}$ (resp., $ 1< p < \frac{n+4+2a}{n-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided \begin{equation*}\label{sysdim00} n < 2 + 2 (\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}} + \sqrt{ \frac{pq(q+1)}{p+1} - \sqrt{\frac{pq(q+1)}{p+1}}}). \end{equation*}
Citation: Mostafa Fazly, Nassif Ghoussoub. On the Hénon-Lane-Emden conjecture. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2513-2533. doi: 10.3934/dcds.2014.34.2513
References:
[1]

S. N. Amstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle,, Comm. Partial Differential Equations, 36 (2011), 2011. doi: 10.1080/03605302.2010.534523.

[2]

M. F. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems,, Adv. Differential Equations, 15 (2010), 1033.

[3]

L. A. 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. doi: 10.1002/cpa.3160420304.

[4]

W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.

[5]

C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems,, Nonlinearity, 26 (2013), 2357. doi: 10.1088/0951-7715/26/8/2357.

[6]

C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights,, Proc. Amer. Math. Soc., 140 (2012), 2003. doi: 10.1090/S0002-9939-2011-11351-0.

[7]

E. N. Dancer, Y. Du and Z. M. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent,, J. Diff. Equ., 250 (2011), 3281. doi: 10.1016/j.jde.2011.02.005.

[8]

J. Davila, L. Dupaigne, K. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem,, preprint, (2013).

[9]

Y. Du and Z. Guo, Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations,, Adv. Differential Equations, 18 (2013), 737.

[10]

P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity,, Comm. Pure Appl. Math., 60 (2007), 1731. doi: 10.1002/cpa.20189.

[11]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS,, Courant Lecture Notes in Mathematics, (2010).

[12]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbbR^N$,, J. Math. Pures Appl. (9), 87 (2007), 537. doi: 10.1016/j.matpur.2007.03.001.

[13]

M. Fazly, Liouville type theorems for stable solutions of certain elliptic systems,, Advanced Nonlinear Studies, 12 (2012), 1.

[14]

M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calc. Var. Partial Differential Equations, 47 (2013), 809. doi: 10.1007/s00526-012-0536-x.

[15]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$,, in Mathematical Analysis and Applications, (1981), 369.

[16]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Commun. Pure Appl. Math., 34 (1981), 525. doi: 10.1002/cpa.3160340406.

[17]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196.

[18]

W. Jeong and Y. Lee, Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential,, Nonlinear Analysis, 87 (2013), 126. doi: 10.1016/j.na.2013.04.007.

[19]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^N$,, Comment. Math. Helv., 73 (1998), 206. doi: 10.1007/s000140050052.

[20]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^N$,, Differential Integral Equations, 9 (1996), 465.

[21]

E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equations, 18 (1993), 125. doi: 10.1080/03605309308820923.

[22]

E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,, Tr. Mat. Inst. Steklova, 234 (2001), 1.

[23]

Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems,, Adv. Diff. Equ., 17 (2012), 605.

[24]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations,, J. Diff. Equ., 252 (2012), 2544. doi: 10.1016/j.jde.2011.09.022.

[25]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, Duke Math. J., 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[26]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.

[27]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).

[28]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, Differential Integral Equations, 9 (1996), 635.

[29]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369.

[30]

Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, Adv. Math., 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014.

[31]

M. A. S. Souto, A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems,, Differential Integral Equations, 8 (1995), 1245.

[32]

C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations,, J. Func. Anal., 262 (2012), 1705. doi: 10.1016/j.jfa.2011.11.017.

[33]

J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem,, Mathematische Annalen, 356 (2013), 1599. doi: 10.1007/s00208-012-0894-x.

show all references

References:
[1]

S. N. Amstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle,, Comm. Partial Differential Equations, 36 (2011), 2011. doi: 10.1080/03605302.2010.534523.

[2]

M. F. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems,, Adv. Differential Equations, 15 (2010), 1033.

[3]

L. A. 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. doi: 10.1002/cpa.3160420304.

[4]

W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.

[5]

C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems,, Nonlinearity, 26 (2013), 2357. doi: 10.1088/0951-7715/26/8/2357.

[6]

C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights,, Proc. Amer. Math. Soc., 140 (2012), 2003. doi: 10.1090/S0002-9939-2011-11351-0.

[7]

E. N. Dancer, Y. Du and Z. M. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent,, J. Diff. Equ., 250 (2011), 3281. doi: 10.1016/j.jde.2011.02.005.

[8]

J. Davila, L. Dupaigne, K. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem,, preprint, (2013).

[9]

Y. Du and Z. Guo, Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations,, Adv. Differential Equations, 18 (2013), 737.

[10]

P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity,, Comm. Pure Appl. Math., 60 (2007), 1731. doi: 10.1002/cpa.20189.

[11]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS,, Courant Lecture Notes in Mathematics, (2010).

[12]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbbR^N$,, J. Math. Pures Appl. (9), 87 (2007), 537. doi: 10.1016/j.matpur.2007.03.001.

[13]

M. Fazly, Liouville type theorems for stable solutions of certain elliptic systems,, Advanced Nonlinear Studies, 12 (2012), 1.

[14]

M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calc. Var. Partial Differential Equations, 47 (2013), 809. doi: 10.1007/s00526-012-0536-x.

[15]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$,, in Mathematical Analysis and Applications, (1981), 369.

[16]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Commun. Pure Appl. Math., 34 (1981), 525. doi: 10.1002/cpa.3160340406.

[17]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196.

[18]

W. Jeong and Y. Lee, Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential,, Nonlinear Analysis, 87 (2013), 126. doi: 10.1016/j.na.2013.04.007.

[19]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^N$,, Comment. Math. Helv., 73 (1998), 206. doi: 10.1007/s000140050052.

[20]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^N$,, Differential Integral Equations, 9 (1996), 465.

[21]

E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equations, 18 (1993), 125. doi: 10.1080/03605309308820923.

[22]

E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,, Tr. Mat. Inst. Steklova, 234 (2001), 1.

[23]

Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems,, Adv. Diff. Equ., 17 (2012), 605.

[24]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations,, J. Diff. Equ., 252 (2012), 2544. doi: 10.1016/j.jde.2011.09.022.

[25]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, Duke Math. J., 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[26]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.

[27]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).

[28]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, Differential Integral Equations, 9 (1996), 635.

[29]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369.

[30]

Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, Adv. Math., 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014.

[31]

M. A. S. Souto, A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems,, Differential Integral Equations, 8 (1995), 1245.

[32]

C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations,, J. Func. Anal., 262 (2012), 1705. doi: 10.1016/j.jfa.2011.11.017.

[33]

J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem,, Mathematische Annalen, 356 (2013), 1599. doi: 10.1007/s00208-012-0894-x.

[1]

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

[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]

Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809

[4]

Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167

[5]

Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793

[6]

Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510

[7]

Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915

[8]

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

[9]

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

[10]

Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094

[11]

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

[12]

Junichi Harada, Mitsuharu Ôtani. $H^2$-solutions for some elliptic equations with nonlinear boundary conditions. Conference Publications, 2009, 2009 (Special) : 333-339. doi: 10.3934/proc.2009.2009.333

[13]

Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure & Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785

[14]

Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107

[15]

Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823

[16]

Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729

[17]

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

[18]

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

[19]

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

[20]

Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure & Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]