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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.
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