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Method of sub-super solutions for fractional elliptic equations
1. | School of Mathematics, Hunan University, Changsha 410082, Hunan, China |
2. | School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522, NSW, Australia |
3. | School of Mathematics(Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China |
$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u),&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right.$ |
$f:\Omega \text{ }\!\!\times\!\!\text{ }\mathbb{R}\to \mathbb{R}$ |
$ν$ |
$ \left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u)+\nu ,&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right. $ |
References:
[1] |
N. Abatangelo,
Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.
doi: 10.3934/dcds.2015.35.5555. |
[2] |
K. Akô,
On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.
doi: 10.2969/jmsj/01310045. |
[3] |
C. Brandle, E. Colorado, A. Pablo and U. Sanchez,
A concave convex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh Sect. A, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[4] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[5] |
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. |
[6] |
H. Chen, P. Felmer and A. Quass,
Large solutions to elliptic equations involving the fractional Laplacian, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 32 (2015), 1199-1228.
doi: 10.1016/j.anihpc.2014.08.001. |
[7] |
H. Chen and L. Véron,
Semilinear fractional elliptic equations involving measures, J. Differential Equations, 257 (2014), 1457-1486.
doi: 10.1016/j.jde.2014.05.012. |
[8] |
H. Chen and L. Véron,
Semilinear fractional elliptic equations with gradient nonlinearity involving measures, J. Funct. Anal., 266 (2014), 5467-5492.
doi: 10.1016/j.jfa.2013.11.009. |
[9] |
W. Chen, L. Ambrosio and Y. Li,
Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381.
doi: 10.1016/j.na.2014.11.003. |
[10] |
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. |
[11] |
W. Chen and J. Y. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[12] |
P. Clément and G. Sweer,
Getting a solution between sub-and suprsolutions without monotone iteration, Rend, Istit. Mat. Univ. Trieste, 19 (1987), 189-194.
|
[13] |
E. N. Dancer and G. Sweer,
On the existence of a maximal weak solution for a semilinear elliptic equation, Differential Integral Equations, 2 (1989), 533-540.
|
[14] |
M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic
problems in the half space, Commun. Contemp. Math. , 18 (2016), 1550012, 25pp.
doi: 10.1142/S0219199715500121. |
[15] |
P. Felmer and A. Quass,
Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.
|
[16] |
M. Montenegro and A. C. Ponce,
The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429-2438.
doi: 10.1090/S0002-9939-08-09231-9. |
[17] |
X. Rosoton 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. |
[18] |
L. Silvestre,
Regularity of the obstacle problem for the fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
show all references
References:
[1] |
N. Abatangelo,
Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.
doi: 10.3934/dcds.2015.35.5555. |
[2] |
K. Akô,
On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.
doi: 10.2969/jmsj/01310045. |
[3] |
C. Brandle, E. Colorado, A. Pablo and U. Sanchez,
A concave convex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh Sect. A, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[4] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[5] |
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. |
[6] |
H. Chen, P. Felmer and A. Quass,
Large solutions to elliptic equations involving the fractional Laplacian, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 32 (2015), 1199-1228.
doi: 10.1016/j.anihpc.2014.08.001. |
[7] |
H. Chen and L. Véron,
Semilinear fractional elliptic equations involving measures, J. Differential Equations, 257 (2014), 1457-1486.
doi: 10.1016/j.jde.2014.05.012. |
[8] |
H. Chen and L. Véron,
Semilinear fractional elliptic equations with gradient nonlinearity involving measures, J. Funct. Anal., 266 (2014), 5467-5492.
doi: 10.1016/j.jfa.2013.11.009. |
[9] |
W. Chen, L. Ambrosio and Y. Li,
Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381.
doi: 10.1016/j.na.2014.11.003. |
[10] |
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. |
[11] |
W. Chen and J. Y. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[12] |
P. Clément and G. Sweer,
Getting a solution between sub-and suprsolutions without monotone iteration, Rend, Istit. Mat. Univ. Trieste, 19 (1987), 189-194.
|
[13] |
E. N. Dancer and G. Sweer,
On the existence of a maximal weak solution for a semilinear elliptic equation, Differential Integral Equations, 2 (1989), 533-540.
|
[14] |
M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic
problems in the half space, Commun. Contemp. Math. , 18 (2016), 1550012, 25pp.
doi: 10.1142/S0219199715500121. |
[15] |
P. Felmer and A. Quass,
Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.
|
[16] |
M. Montenegro and A. C. Ponce,
The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429-2438.
doi: 10.1090/S0002-9939-08-09231-9. |
[17] |
X. Rosoton 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. |
[18] |
L. Silvestre,
Regularity of the obstacle problem for the fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
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