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Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model
Qualitative properties of positive solutions for mixed integro-differential equations
1. | Departamento de Ingeniería Matemática and Centro de Modelamiento, Matemático, Universidad de Chile, Santiago, Chile |
2. | Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
$\begin{equation}\left\{ \begin{array}{l} (-\Delta)_x^{α} u+(-\Delta)_y u+u = f(u) \ \ \ \ {\rm in}\ \ {\mathbb{R}}^N×{\mathbb{R}}^M,\\ u>0\ \ {\rm{in}}\ {\mathbb{R}}^N\times{\mathbb{R}}^M,\ \ \ \lim_{|(x,y)|\to+\infty}u(x,y) = 0, \end{array} \right.\;\;\;\left( {0.1} \right)\end{equation}$ |
$N≥ 1$ |
$M≥ 1$ |
$α∈ (0,1)$ |
References:
[1] |
G. Barles, R. Buckdahn and E. Pardoux,
Backward stochastic differential equations and integral partial differential equations, Stochastics Stochastics Rep., 60 (1997), 57-83.
doi: 10.1080/17442509708834099. |
[2] |
G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert,
Lipschitz regularity of solutions for mixed integro-differential equations, J. Diff. Eq., 252 (2012), 6012-6060.
doi: 10.1016/j.jde.2012.02.013. |
[3] |
G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert,
Large time behavior of periodic viscosity solutions for uniformly elliptic integro-differential equations, Calc. Var. Part. Diff. Eq., 50 (2014), 283-304.
doi: 10.1007/s00526-013-0636-2. |
[4] |
F. Benth, K. Karlsen and K. Reikvam,
Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach, Finance Stoch., 5 (2001), 275-303.
doi: 10.1007/PL00013538. |
[5] |
H. Berestycki and P. L. Lions,
Non linear scalar field equations Ⅰ: Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[6] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasileira Mat., 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[7] |
F. Brock,
Continuous steiner-symmetrization, Mathematische Nachri- chten, 172 (1995), 25-48.
doi: 10.1002/mana.19951720104. |
[8] |
J. Busca and P. Felmer,
Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. Partial Differential Equations, 13 (2001), 191-211.
doi: 10.1007/PL00009928. |
[9] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2014), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[10] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
A. Ciomaga,
On the strong maximum principle for second order nonlinear parabolic integro-differential equations, Adv. Diff. Eq., 17 (2012), 635-671.
|
[13] |
C. Cortázar, M. Elgueta and P. Felmer,
On a semilinear elliptic problem in ${\mathbb{R}}^N$ with a non-lipschitzian non-linearity, Adv. Diff. Eq., 1 (1996), 199-218.
|
[14] |
F. Da Lio and B. Sirakov,
Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc., 9 (2007), 317-330.
doi: 10.4171/JEMS/81. |
[15] |
S. Dipierro, G. Palatucci and E. Valdinoci,
Existence and symmetry results for a schrödinger type problem involving the fractional laplacian, Le Matematiche (Catania), 68 (2013), 201-206.
|
[16] |
J. Dolbeault and P. Felmer,
Symmetry and monotonicity properties for positive solution of semi-linear elliptic PDE's, Comm. Part. Diff. Eq., 25 (2000), 1153-1169.
doi: 10.1080/03605300008821545. |
[17] |
A. Esfahani and S. E. Esfahani, Positive and nodal solutions of the generalized BO-ZK equation, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A. Matematicas, DOI: 10.1007/s13398-017-0435-2(2017). |
[18] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of non-linear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[19] |
P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Contem. Math., 16 (2014), 1350023, 24 pp.
doi: 10.1142/S0219199713500235. |
[20] |
B. Gidas, W. M. Ni and L. Nirenberg,
Symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Math. Anal. Appl., Part A, Advances in Math. Suppl. Studied, 7A (1981), 369-402.
|
[21] |
C. Gui,
Symmetry of the blow up set of a porous medium type equation, Comm. Pure Appl. Math., 48 (1995), 471-500.
doi: 10.1002/cpa.3160480502. |
[22] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\mathbb{R}}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[23] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. |
[24] |
C. M. Li,
Monotonicity and symmetry of solutions of fully non-linear elliptic equations on unbounded domains, Comm. Part. Diff. Eq., 16 (1991), 585-615.
doi: 10.1080/03605309108820770. |
[25] |
Y. Li and W. M. Ni,
Radial symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Comm. Part. Diff. Eq., 18 (1993), 1043-1054.
doi: 10.1080/03605309308820960. |
[26] |
Y. Y. Li,
Remark on some conformally invariant integral equations: The method fo moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
[27] |
F. Pacella and M. Ramaswamy,
Symmetry of solutions of elliptic equations via maximum principles, Handbook of Differential Equations (M. Chipot, ed.), Elsevier, 4 (2008), 269-312.
doi: 10.1016/S1874-5733(08)80021-6. |
[28] |
H. Pham,
Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems Estim. Control, 8 (1998), 1-27.
|
[29] |
A. Quaas and A. Xia,
Liouville type theorems for nonlinear elliptic equations and systems involving fractional laplacian in the half space, Calc. Var. Part. Diff. Eq., 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
[30] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[31] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[32] |
R. Servadei and E. Valdinoci,
A Brezis-Nirenberg result for non-local critical equations in low dimension, Comm. Pure Appl. Anal., 12 (2013), 2445-2464.
doi: 10.3934/cpaa.2013.12.2445. |
[33] |
Y. Sire and E. Valdinoci,
Fractional laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
show all references
References:
[1] |
G. Barles, R. Buckdahn and E. Pardoux,
Backward stochastic differential equations and integral partial differential equations, Stochastics Stochastics Rep., 60 (1997), 57-83.
doi: 10.1080/17442509708834099. |
[2] |
G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert,
Lipschitz regularity of solutions for mixed integro-differential equations, J. Diff. Eq., 252 (2012), 6012-6060.
doi: 10.1016/j.jde.2012.02.013. |
[3] |
G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert,
Large time behavior of periodic viscosity solutions for uniformly elliptic integro-differential equations, Calc. Var. Part. Diff. Eq., 50 (2014), 283-304.
doi: 10.1007/s00526-013-0636-2. |
[4] |
F. Benth, K. Karlsen and K. Reikvam,
Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach, Finance Stoch., 5 (2001), 275-303.
doi: 10.1007/PL00013538. |
[5] |
H. Berestycki and P. L. Lions,
Non linear scalar field equations Ⅰ: Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[6] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasileira Mat., 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[7] |
F. Brock,
Continuous steiner-symmetrization, Mathematische Nachri- chten, 172 (1995), 25-48.
doi: 10.1002/mana.19951720104. |
[8] |
J. Busca and P. Felmer,
Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. Partial Differential Equations, 13 (2001), 191-211.
doi: 10.1007/PL00009928. |
[9] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2014), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[10] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
A. Ciomaga,
On the strong maximum principle for second order nonlinear parabolic integro-differential equations, Adv. Diff. Eq., 17 (2012), 635-671.
|
[13] |
C. Cortázar, M. Elgueta and P. Felmer,
On a semilinear elliptic problem in ${\mathbb{R}}^N$ with a non-lipschitzian non-linearity, Adv. Diff. Eq., 1 (1996), 199-218.
|
[14] |
F. Da Lio and B. Sirakov,
Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc., 9 (2007), 317-330.
doi: 10.4171/JEMS/81. |
[15] |
S. Dipierro, G. Palatucci and E. Valdinoci,
Existence and symmetry results for a schrödinger type problem involving the fractional laplacian, Le Matematiche (Catania), 68 (2013), 201-206.
|
[16] |
J. Dolbeault and P. Felmer,
Symmetry and monotonicity properties for positive solution of semi-linear elliptic PDE's, Comm. Part. Diff. Eq., 25 (2000), 1153-1169.
doi: 10.1080/03605300008821545. |
[17] |
A. Esfahani and S. E. Esfahani, Positive and nodal solutions of the generalized BO-ZK equation, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A. Matematicas, DOI: 10.1007/s13398-017-0435-2(2017). |
[18] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of non-linear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[19] |
P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Contem. Math., 16 (2014), 1350023, 24 pp.
doi: 10.1142/S0219199713500235. |
[20] |
B. Gidas, W. M. Ni and L. Nirenberg,
Symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Math. Anal. Appl., Part A, Advances in Math. Suppl. Studied, 7A (1981), 369-402.
|
[21] |
C. Gui,
Symmetry of the blow up set of a porous medium type equation, Comm. Pure Appl. Math., 48 (1995), 471-500.
doi: 10.1002/cpa.3160480502. |
[22] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\mathbb{R}}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[23] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. |
[24] |
C. M. Li,
Monotonicity and symmetry of solutions of fully non-linear elliptic equations on unbounded domains, Comm. Part. Diff. Eq., 16 (1991), 585-615.
doi: 10.1080/03605309108820770. |
[25] |
Y. Li and W. M. Ni,
Radial symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Comm. Part. Diff. Eq., 18 (1993), 1043-1054.
doi: 10.1080/03605309308820960. |
[26] |
Y. Y. Li,
Remark on some conformally invariant integral equations: The method fo moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
[27] |
F. Pacella and M. Ramaswamy,
Symmetry of solutions of elliptic equations via maximum principles, Handbook of Differential Equations (M. Chipot, ed.), Elsevier, 4 (2008), 269-312.
doi: 10.1016/S1874-5733(08)80021-6. |
[28] |
H. Pham,
Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems Estim. Control, 8 (1998), 1-27.
|
[29] |
A. Quaas and A. Xia,
Liouville type theorems for nonlinear elliptic equations and systems involving fractional laplacian in the half space, Calc. Var. Part. Diff. Eq., 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
[30] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[31] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[32] |
R. Servadei and E. Valdinoci,
A Brezis-Nirenberg result for non-local critical equations in low dimension, Comm. Pure Appl. Anal., 12 (2013), 2445-2464.
doi: 10.3934/cpaa.2013.12.2445. |
[33] |
Y. Sire and E. Valdinoci,
Fractional laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
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