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A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities
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Admissibility and polynomial dichotomies for evolution families
Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation
1. | School of mathematics and statistics, Anhui normal university, Wuhu, 241002, China |
2. | School of Information and Mathematics, Yangtze University, Jingzhou 434023, China |
$ \begin{cases} F_\alpha(u)(x) = f(x,u,\nabla u) \ \mbox{in} \ \Omega,\\ u>0, \ \mbox{in}\ \Omega; \ u\equiv0, \ \mbox{in}\ \mathbb R^n\backslash\Omega, \end{cases} $ |
$ \Omega $ |
$ \mathbb R^n $ |
$ x_1- $ |
$ x_1- $ |
$ \Omega $ |
$ \mathbb R^n_+ $ |
References:
[1] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[2] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[3] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[4] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[5] |
W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18.
doi: 10.1007/s00526-017-1110-3. |
[6] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[7] |
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. |
[8] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[9] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), 1750018, 12.
doi: 10.1142/S0219199717500183. |
[10] |
Y. Fang and W. Chen,
A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[11] |
R. L. Frank and E. H. Lieb,
Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.
doi: 10.1007/s00526-009-0302-x. |
[12] |
X. Han, G. Lu and J. Zhu,
Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.
doi: 10.1016/j.jde.2011.07.037. |
[13] |
F. Hang,
On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.
doi: 10.4310/MRL.2007.v14.n3.a2. |
[14] |
F. Hang, X. Wang and X. Yan,
An integral equation in conformal geometry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1-21.
doi: 10.1016/j.anihpc.2007.03.006. |
[15] |
Y. Lei,
Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.
doi: 10.1016/j.jde.2012.11.008. |
[16] |
C. Li, Z. Wu and H. Xu,
Maximum principles and bôcher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.
doi: 10.1073/pnas.1804225115. |
[17] |
G. Lu and J. Zhu,
Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577.
doi: 10.1007/s00526-011-0398-7. |
[18] |
G. Lu and J. Zhu,
An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.
doi: 10.1016/j.na.2011.11.036. |
[19] |
P. Niu, L. Wu and X. Ji,
Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fractional Calculus and Applied Analysis, 21 (2018), 552-574.
doi: 10.1515/fca-2018-0030. |
[20] |
Y. Wang and J. Wang,
The method of moving planes for integral equation in an extremal case, J. Partial Differ. Equ., 29 (2016), 246-254.
doi: 10.4208/jpde.v29.n3.6. |
show all references
References:
[1] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[2] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[3] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[4] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[5] |
W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18.
doi: 10.1007/s00526-017-1110-3. |
[6] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[7] |
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. |
[8] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[9] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), 1750018, 12.
doi: 10.1142/S0219199717500183. |
[10] |
Y. Fang and W. Chen,
A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[11] |
R. L. Frank and E. H. Lieb,
Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.
doi: 10.1007/s00526-009-0302-x. |
[12] |
X. Han, G. Lu and J. Zhu,
Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.
doi: 10.1016/j.jde.2011.07.037. |
[13] |
F. Hang,
On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.
doi: 10.4310/MRL.2007.v14.n3.a2. |
[14] |
F. Hang, X. Wang and X. Yan,
An integral equation in conformal geometry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1-21.
doi: 10.1016/j.anihpc.2007.03.006. |
[15] |
Y. Lei,
Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.
doi: 10.1016/j.jde.2012.11.008. |
[16] |
C. Li, Z. Wu and H. Xu,
Maximum principles and bôcher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.
doi: 10.1073/pnas.1804225115. |
[17] |
G. Lu and J. Zhu,
Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577.
doi: 10.1007/s00526-011-0398-7. |
[18] |
G. Lu and J. Zhu,
An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.
doi: 10.1016/j.na.2011.11.036. |
[19] |
P. Niu, L. Wu and X. Ji,
Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fractional Calculus and Applied Analysis, 21 (2018), 552-574.
doi: 10.1515/fca-2018-0030. |
[20] |
Y. Wang and J. Wang,
The method of moving planes for integral equation in an extremal case, J. Partial Differ. Equ., 29 (2016), 246-254.
doi: 10.4208/jpde.v29.n3.6. |
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