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Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$
Liouville theorems for stable solutions of the weighted Lane-Emden system
1. | Institut Préparatoire aux Etudes d'Ingénieurs, Université de Kairouan, Tunisie |
2. | Institut Supérieur des Mathématiques Appliquées et de l'Informatique, Université de Kairouan, Tunisie |
3. | Institut Élie Cartan de Lorraine, IECL, UMR 7502, Université de Lorraine, France |
$-Δ u = ρ(x)v^p, -Δ v= ρ(x)u^θ, u,v>0 \;\mbox{in }\;\mathbb{R}^N$ |
$1 <p≤qθ$ |
$ρ: \mathbb{R}^N \to \mathbb{R}$ |
$ρ(x)≥q A(1+|x|^2)^{\frac{α}{2}}$ |
$\mathbb{R}^N$ |
$α≥q 0$ |
$A>0$ |
$1 < p ≤q \frac{4}{3}$ |
$-Δ u = ρ(x)u^p$ |
$\mathbb{R}^N$ |
References:
[1] |
W. Chen, L. Dupaigne and M. Ghergu,
A new critical curve for the Lane-Emden system, Discrete Contin. Dyn. Syst., 34 (2014), 2469--2479.
doi: 10.3934/dcds.2014.34.2469. |
[2] |
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. |
[3] |
C. Cowan,
Regularity of stable solutions of a Lane-Emden type system, Methods Appl. Anal., 22 (2015), 301-311.
doi: 10.4310/MAA.2015.v22.n3.a4. |
[4] |
C. Cowan and M. Fazly,
On stable entire solutions of semilinear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.
doi: 10.1090/S0002-9939-2011-11351-0. |
[5] |
C. Cowan and N. Ghoussoub,
Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. PDE., 49 (2014), 291-305.
doi: 10.1007/s00526-012-0582-4. |
[6] |
J. Dávila, L. Dupaigne, K. Wang and J. Wei,
A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.
doi: 10.1016/j.aim.2014.02.034. |
[7] |
L. Dupaigne, A. Farina and B. Sirakov,
Regularity of the extremal solutions for the Liouville system, in: Geometric Partial Differential Equations, Publications of the Scuola Normale Superiore/CRM Series, 15 (2013), 139-144.
doi: 10.1007/978-88-7642-473-1_7. |
[8] |
A. Farina,
On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[9] |
M. Fazly,
Liouville type theorems for stable solutions of certain elliptic systems, Adv. Nonlinear Stud., 12 (2012), 1-17.
doi: 10.1515/ans-2012-0101. |
[10] |
C. Gui, W. Ni and X. Wang,
On the stability and instability of positive steady states of a semilinear heat equation in $\textbf{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.
doi: 10.1002/cpa.3160450906. |
[11] |
H. Hajlaoui, A. Harrabi and D. Ye,
On stable solutions of the biharmonic problem with polynomial growth, Pacific J. Math., 270 (2014), 79-93.
doi: 10.2140/pjm.2014.270.79. |
[12] |
L. Hu,
Liouville type results for semi-stable solutions of the weighted Lane-Emden system, J. Math. Anal. Appl., 432 (2015), 429-440.
doi: 10.1016/j.jmaa.2015.06.032. |
[13] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.
|
[14] |
E. Mitidieri and S. Pohozaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.
|
[15] |
M. Montenegro,
Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.
doi: 10.1112/S0024609305004248. |
[16] |
P. Polácik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅰ: Elliptic systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[17] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equations, 9 (1996), 635-653.
|
[18] |
P. 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. |
[19] |
J. Wei and D. Ye,
Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612.
doi: 10.1007/s00208-012-0894-x. |
show all references
References:
[1] |
W. Chen, L. Dupaigne and M. Ghergu,
A new critical curve for the Lane-Emden system, Discrete Contin. Dyn. Syst., 34 (2014), 2469--2479.
doi: 10.3934/dcds.2014.34.2469. |
[2] |
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. |
[3] |
C. Cowan,
Regularity of stable solutions of a Lane-Emden type system, Methods Appl. Anal., 22 (2015), 301-311.
doi: 10.4310/MAA.2015.v22.n3.a4. |
[4] |
C. Cowan and M. Fazly,
On stable entire solutions of semilinear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.
doi: 10.1090/S0002-9939-2011-11351-0. |
[5] |
C. Cowan and N. Ghoussoub,
Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. PDE., 49 (2014), 291-305.
doi: 10.1007/s00526-012-0582-4. |
[6] |
J. Dávila, L. Dupaigne, K. Wang and J. Wei,
A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.
doi: 10.1016/j.aim.2014.02.034. |
[7] |
L. Dupaigne, A. Farina and B. Sirakov,
Regularity of the extremal solutions for the Liouville system, in: Geometric Partial Differential Equations, Publications of the Scuola Normale Superiore/CRM Series, 15 (2013), 139-144.
doi: 10.1007/978-88-7642-473-1_7. |
[8] |
A. Farina,
On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[9] |
M. Fazly,
Liouville type theorems for stable solutions of certain elliptic systems, Adv. Nonlinear Stud., 12 (2012), 1-17.
doi: 10.1515/ans-2012-0101. |
[10] |
C. Gui, W. Ni and X. Wang,
On the stability and instability of positive steady states of a semilinear heat equation in $\textbf{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.
doi: 10.1002/cpa.3160450906. |
[11] |
H. Hajlaoui, A. Harrabi and D. Ye,
On stable solutions of the biharmonic problem with polynomial growth, Pacific J. Math., 270 (2014), 79-93.
doi: 10.2140/pjm.2014.270.79. |
[12] |
L. Hu,
Liouville type results for semi-stable solutions of the weighted Lane-Emden system, J. Math. Anal. Appl., 432 (2015), 429-440.
doi: 10.1016/j.jmaa.2015.06.032. |
[13] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.
|
[14] |
E. Mitidieri and S. Pohozaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.
|
[15] |
M. Montenegro,
Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.
doi: 10.1112/S0024609305004248. |
[16] |
P. Polácik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅰ: Elliptic systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[17] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equations, 9 (1996), 635-653.
|
[18] |
P. 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. |
[19] |
J. Wei and D. Ye,
Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612.
doi: 10.1007/s00208-012-0894-x. |
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