On the Hénon-Lane-Emden conjecture

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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

Abstract
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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 - 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.

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