American Institute of Mathematical Sciences

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

Method of sub-super solutions for fractional elliptic equations

Yanqin Fang 1,2, and  De Tang 3,,

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.
Keywords: Fractional Laplacian, Radon measure, Caratheódory function, subsolution, supersolution.
Mathematics Subject Classification: Primary: 35J60, 35J67; Secondary: 58J05.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

P. Felmer and A. Quass, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

P. Felmer and A. Quass, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017
[2]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741
[3]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
[4]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008
[5]

Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521
[6]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
[7]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076
[8]

Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
[9]

Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[10]

De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431
[11]

Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046
[12]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
[13]

Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209
[14]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
[15]

Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813
[16]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020268
[17]

Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068
[18]

Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236
[19]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[20]

David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (218)
  • HTML views (608)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences