• Previous Article
    The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds
  • DCDS Home
  • This Issue
  • Next Article
    Some symmetry results for entire solutions of an elliptic system arising in phase separation
June  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, 2014, 34 (6) : 2513-2533. doi: 10.3934/dcds.2014.34.2513
References:
[1]

Comm. Partial Differential Equations, 36 (2011), 2011-2047. doi: 10.1080/03605302.2010.534523.  Google Scholar

[2]

Adv. Differential Equations, 15 (2010), 1033-1082.  Google Scholar

[3]

Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[5]

Nonlinearity, 26 (2013), 2357-2371. doi: 10.1088/0951-7715/26/8/2357.  Google Scholar

[6]

Proc. Amer. Math. Soc., 140 (2012), 2003-2012. doi: 10.1090/S0002-9939-2011-11351-0.  Google Scholar

[7]

J. Diff. Equ., 250 (2011), 3281-3310. doi: 10.1016/j.jde.2011.02.005.  Google Scholar

[8]

preprint, 2013. Google Scholar

[9]

Adv. Differential Equations, 18 (2013), 737-768.  Google Scholar

[10]

Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189.  Google Scholar

[11]

Courant Lecture Notes in Mathematics, 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010.  Google Scholar

[12]

J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[13]

Advanced Nonlinear Studies, 12 (2012), 1-17.  Google Scholar

[14]

Calc. Var. Partial Differential Equations, 47 (2013), 809-823. doi: 10.1007/s00526-012-0536-x.  Google Scholar

[15]

in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[16]

Commun. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[17]

Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[18]

Nonlinear Analysis, 87 (2013), 126-145. doi: 10.1016/j.na.2013.04.007.  Google Scholar

[19]

Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.  Google Scholar

[20]

Differential Integral Equations, 9 (1996), 465-479.  Google Scholar

[21]

Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[22]

Tr. Mat. Inst. Steklova, 234 (2001), 1-384.  Google Scholar

[23]

Adv. Diff. Equ., 17 (2012), 605-634.  Google Scholar

[24]

J. Diff. Equ., 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[25]

Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[26]

Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[27]

Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007.  Google Scholar

[28]

Differential Integral Equations, 9 (1996), 635-653.  Google Scholar

[29]

Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

[30]

Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[31]

Differential Integral Equations, 8 (1995), 1245-1258.  Google Scholar

[32]

J. Func. Anal., 262 (2012), 1705-1727. doi: 10.1016/j.jfa.2011.11.017.  Google Scholar

[33]

Mathematische Annalen, 356 (2013), 1599-1612. doi: 10.1007/s00208-012-0894-x.  Google Scholar

show all references

References:
[1]

Comm. Partial Differential Equations, 36 (2011), 2011-2047. doi: 10.1080/03605302.2010.534523.  Google Scholar

[2]

Adv. Differential Equations, 15 (2010), 1033-1082.  Google Scholar

[3]

Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[5]

Nonlinearity, 26 (2013), 2357-2371. doi: 10.1088/0951-7715/26/8/2357.  Google Scholar

[6]

Proc. Amer. Math. Soc., 140 (2012), 2003-2012. doi: 10.1090/S0002-9939-2011-11351-0.  Google Scholar

[7]

J. Diff. Equ., 250 (2011), 3281-3310. doi: 10.1016/j.jde.2011.02.005.  Google Scholar

[8]

preprint, 2013. Google Scholar

[9]

Adv. Differential Equations, 18 (2013), 737-768.  Google Scholar

[10]

Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189.  Google Scholar

[11]

Courant Lecture Notes in Mathematics, 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010.  Google Scholar

[12]

J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[13]

Advanced Nonlinear Studies, 12 (2012), 1-17.  Google Scholar

[14]

Calc. Var. Partial Differential Equations, 47 (2013), 809-823. doi: 10.1007/s00526-012-0536-x.  Google Scholar

[15]

in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[16]

Commun. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[17]

Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[18]

Nonlinear Analysis, 87 (2013), 126-145. doi: 10.1016/j.na.2013.04.007.  Google Scholar

[19]

Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.  Google Scholar

[20]

Differential Integral Equations, 9 (1996), 465-479.  Google Scholar

[21]

Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[22]

Tr. Mat. Inst. Steklova, 234 (2001), 1-384.  Google Scholar

[23]

Adv. Diff. Equ., 17 (2012), 605-634.  Google Scholar

[24]

J. Diff. Equ., 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[25]

Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[26]

Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[27]

Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007.  Google Scholar

[28]

Differential Integral Equations, 9 (1996), 635-653.  Google Scholar

[29]

Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

[30]

Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[31]

Differential Integral Equations, 8 (1995), 1245-1258.  Google Scholar

[32]

J. Func. Anal., 262 (2012), 1705-1727. doi: 10.1016/j.jfa.2011.11.017.  Google Scholar

[33]

Mathematische Annalen, 356 (2013), 1599-1612. doi: 10.1007/s00208-012-0894-x.  Google Scholar

[1]

Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036

[2]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016

[3]

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033

[4]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019

[5]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[6]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[7]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2653-2676. doi: 10.3934/dcds.2020379

[8]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3629-3650. doi: 10.3934/dcds.2021010

[9]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[10]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[11]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[12]

Yu-Hsien Liao. Solutions and characterizations under multicriteria management systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021041

[13]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211

[14]

Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021066

[15]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[16]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[17]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241

[18]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[19]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058

[20]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]