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June  2014, 34(6): 2513-2533. doi: 10.3934/dcds.2014.34.2513

On the Hénon-Lane-Emden conjecture

Mostafa Fazly 1, and  Nassif Ghoussoub 2,

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*}
Keywords: stable solutions, Liouville theorems, nonlinear elliptic systems, finite Morse index solutions, Hénon-Lane-Emden conjecture..
Mathematics Subject Classification: 35J47, 35B33, 35B45, 35B0.
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
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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

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