American Institute of Mathematical Sciences

• 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 and Continuous Dynamical Systems, 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), 2011-2047. 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), 1033-1082. [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), 271-297. 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), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [5] C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2013), 2357-2371. 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), 2003-2012. 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), 3281-3310. 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), 737-768. [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), 1731-1768. 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, 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 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), 537-561. 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), 1-17. [14] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calc. Var. Partial Differential Equations, 47 (2013), 809-823. 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, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. [16] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598. 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), 883-901. 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), 126-145. 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), 206-231. doi: 10.1007/s000140050052. [20] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^N$, Differential Integral Equations, 9 (1996), 465-479. [21] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. 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), 1-384. [23] Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems, Adv. Diff. Equ., 17 (2012), 605-634. [24] Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Diff. Equ., 252 (2012), 2544-2562. 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), 555-579. doi: 10.1215/S0012-7094-07-13935-8. [26] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. 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], Birkhäuser Verlag, Basel, 2007. [28] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. [29] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380. [30] Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427. 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), 1245-1258. [32] C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Func. Anal., 262 (2012), 1705-1727. 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), 1599-1612. 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), 2011-2047. 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), 1033-1082. [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), 271-297. 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), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [5] C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2013), 2357-2371. 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), 2003-2012. 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), 3281-3310. 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), 737-768. [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), 1731-1768. 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, 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 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), 537-561. 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), 1-17. [14] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calc. Var. Partial Differential Equations, 47 (2013), 809-823. 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, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. [16] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598. 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), 883-901. 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), 126-145. 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), 206-231. doi: 10.1007/s000140050052. [20] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^N$, Differential Integral Equations, 9 (1996), 465-479. [21] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. 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), 1-384. [23] Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems, Adv. Diff. Equ., 17 (2012), 605-634. [24] Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Diff. Equ., 252 (2012), 2544-2562. 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), 555-579. doi: 10.1215/S0012-7094-07-13935-8. [26] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. 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], Birkhäuser Verlag, Basel, 2007. [28] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. [29] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380. [30] Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427. 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), 1245-1258. [32] C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Func. Anal., 262 (2012), 1705-1727. 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), 1599-1612. 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 and 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 and Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011 [3] Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062 [4] Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081 [5] Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003 [6] Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure and Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025 [7] Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4185-4206. doi: 10.3934/dcds.2021033 [8] Foued Mtiri. Liouville type theorems for stable solutions of elliptic system involving the Grushin operator. Communications on Pure and Applied Analysis, 2022, 21 (2) : 541-553. doi: 10.3934/cpaa.2021187 [9] Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291 [10] Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809 [11] Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167 [12] Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058 [13] Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793 [14] Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1681-1698. doi: 10.3934/cpaa.2021036 [15] Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510 [16] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [17] Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915 [18] Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869 [19] Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094 [20] M. Á. Burgos-Pérez, J. García-Melián, A. Quaas. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4703-4721. doi: 10.3934/dcds.2016004

2020 Impact Factor: 1.392