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Some symmetry results for entire solutions of an elliptic system arising in phase separation
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 |
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*}
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 $\mathbb{R}^N2$, 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 $\mathbb{R}^N2$, 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 $\mathbb{R}^N2$, Comment. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[20] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^N2$, 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 $\mathbb{R}^N2$, 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 $\mathbb{R}^N2$, 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 $\mathbb{R}^N2$, Comment. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[20] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^N2$, 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. |
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