# American Institute of Mathematical Sciences

October 2018, 23(8): 3153-3165. doi: 10.3934/dcdsb.2017212

## 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

* Corresponding author: tangdehnu@126.com

Received  April 2017 Revised  August 2017 Published  September 2017

Fund Project: The first author is supported by National Natural Sciences Foundations of China 11301166 and Young Teachers Program of Hunan University
The second author is supported by Natural Science Foundation of Hunan Province, China 2016JJ2018

By applying the method of sub-super solutions, we obtain the existence of weak solutions to fractional Laplacian
 $\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u),&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right.$
where
 $f:\Omega \text{ }\!\!\times\!\!\text{ }\mathbb{R}\to \mathbb{R}$
is a Caratheódory function.
Let
 $ν$
be a Radon measure. Based on the existence result in (1), we derive the existence of weak solutions for the semilinear fractional elliptic equation with measure data
 $\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u)+\nu ,&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right.$
Some results in[7] are extended.
In addition, we generalize some results to systems of fractional Laplacian equations by constructing subsolutions and supersolutions.
Citation: Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
##### 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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##### 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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