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January  2022, 21(1): 83-100. doi: 10.3934/cpaa.2021168

Regularity and existence of positive solutions for a fractional system

1. 

Department of Mathematics and Statistics, Huanghuai University, Zhumadian, Henan 463000, China

2. 

Department of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

* Corresponding author

Received  May 2021 Revised  August 2021 Published  January 2022 Early access  September 2021

Fund Project: The first author is partially supported by NSFC 11701207

We consider the nonlinear fractional elliptic system
$ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), & \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), & \text{in}\, \, \, \Omega, \\ u = v = 0, & \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $
where
$ 0<\alpha_1, \alpha_2<2 $
and
$ \Omega $
is a bounded domain with
$ C^2 $
boundary in
$ \mathbb{R}^n $
. To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for
$ 0<\alpha_1, \alpha_2<1 $
and
$ 1<\alpha_1, \alpha_2 <2 $
respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.
Citation: Ran Zhuo, Yan Li. Regularity and existence of positive solutions for a fractional system. Communications on Pure and Applied Analysis, 2022, 21 (1) : 83-100. doi: 10.3934/cpaa.2021168
References:
[1] R. Adams and J. Fournier, Sobolev Spaces, Academic Press, 2003. 
[2]

G. AlbertiG. Bouchitté and P. Seppecher, Phase transition with the line-tension effect, Arch. Ration. Mech. Anal., 144 (1998), 1-46.  doi: 10.1007/s002050050111.

[3]

B. BarriosL. M. Del PezzoJ. G. Melián and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, Rev. Mat. Iberoam., 34 (2018), 195-220.  doi: 10.4171/RMI/983.

[4]

T. BartschN. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.

[5] J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Cambridge Univ. Press, Cambridge, 1996. 
[6]

K. BogdanT. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. 

[7]

L. CaffarelliJ. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.

[8]

L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.  doi: 10.1007/s00205-010-0336-4.

[9]

W. ChenC. 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.

[10]

W. ChenC. Li and Y. Li, A direct blowng-up and rescaling argument on nonlocal elliptic equations, Int. J. Math., 27 (2016), 1650064.  doi: 10.1142/S0129167X16500646.

[11]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd, 2020. doi: 10.1142/10550.

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC Financ. Math. Ser., Chapman and Hall/CRC, Boca Raton, FL, 2004.

[13]

S. Dipierro and A. Pinamont, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differ. Equ., 255 (2013), 85-119.  doi: 10.1016/j.jde.2013.04.001.

[14]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin, 1976, translated from French by C.W. John.

[15]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738.  doi: 10.1016/j.aim.2010.09.023.

[16]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial. Differ. Equ., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[17]

M. d. M. González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differ. Equ., 36 (2009), 173-210.  doi: 10.1007/s00526-009-0225-6.

[18]

D. Kriventsov, $C^{1, \alpha}$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Commun. Partial. Differ. Equ., 38 (2013), 2081-2106.  doi: 10.1080/03605302.2013.831990.

[19]

E. Leite and M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, Topol. Methods Nonlinear Anal., 53 (2019), 407-425.  doi: 10.12775/tmna.2019.005.

[20]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.  doi: 10.1007/s11425-016-0231-x.

[21]

L. Lin, A priori bounds and existence result of positive solutions for fractional Laplacian systems, Discrete Contin. Dyn. Syst., 39 (2019), 1517-1531.  doi: 10.3934/dcds.2019065.

[22]

P. PoláčikP. Quittner and P. Souplet, Singularity and decay estimate in superlinear problems via Liouville-type theorem. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.

[23]

A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equation and systems involving frctional Laplacian in the half space, Calc. Var. Partial Differ. Equ., 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8.

[24]

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.

[25]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[26]

R. Zhuo and Y. Li, Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian, Discrete Contin. Dyn. Syst., 39 (2019), 1595-1611.  doi: 10.3934/dcds.2019071.

[27]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

[28]

A. ZoiaA. Rosso and K. Kardar, Fractional Laplacian in bounded domains, Phys. Rev. E, 76 (2007), 021116.  doi: 10.1103/PhysRevE.76.021116.

show all references

References:
[1] R. Adams and J. Fournier, Sobolev Spaces, Academic Press, 2003. 
[2]

G. AlbertiG. Bouchitté and P. Seppecher, Phase transition with the line-tension effect, Arch. Ration. Mech. Anal., 144 (1998), 1-46.  doi: 10.1007/s002050050111.

[3]

B. BarriosL. M. Del PezzoJ. G. Melián and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, Rev. Mat. Iberoam., 34 (2018), 195-220.  doi: 10.4171/RMI/983.

[4]

T. BartschN. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.

[5] J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Cambridge Univ. Press, Cambridge, 1996. 
[6]

K. BogdanT. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. 

[7]

L. CaffarelliJ. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.

[8]

L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.  doi: 10.1007/s00205-010-0336-4.

[9]

W. ChenC. 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.

[10]

W. ChenC. Li and Y. Li, A direct blowng-up and rescaling argument on nonlocal elliptic equations, Int. J. Math., 27 (2016), 1650064.  doi: 10.1142/S0129167X16500646.

[11]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd, 2020. doi: 10.1142/10550.

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC Financ. Math. Ser., Chapman and Hall/CRC, Boca Raton, FL, 2004.

[13]

S. Dipierro and A. Pinamont, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differ. Equ., 255 (2013), 85-119.  doi: 10.1016/j.jde.2013.04.001.

[14]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin, 1976, translated from French by C.W. John.

[15]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738.  doi: 10.1016/j.aim.2010.09.023.

[16]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial. Differ. Equ., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[17]

M. d. M. González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differ. Equ., 36 (2009), 173-210.  doi: 10.1007/s00526-009-0225-6.

[18]

D. Kriventsov, $C^{1, \alpha}$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Commun. Partial. Differ. Equ., 38 (2013), 2081-2106.  doi: 10.1080/03605302.2013.831990.

[19]

E. Leite and M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, Topol. Methods Nonlinear Anal., 53 (2019), 407-425.  doi: 10.12775/tmna.2019.005.

[20]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.  doi: 10.1007/s11425-016-0231-x.

[21]

L. Lin, A priori bounds and existence result of positive solutions for fractional Laplacian systems, Discrete Contin. Dyn. Syst., 39 (2019), 1517-1531.  doi: 10.3934/dcds.2019065.

[22]

P. PoláčikP. Quittner and P. Souplet, Singularity and decay estimate in superlinear problems via Liouville-type theorem. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.

[23]

A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equation and systems involving frctional Laplacian in the half space, Calc. Var. Partial Differ. Equ., 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8.

[24]

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.

[25]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[26]

R. Zhuo and Y. Li, Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian, Discrete Contin. Dyn. Syst., 39 (2019), 1595-1611.  doi: 10.3934/dcds.2019071.

[27]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

[28]

A. ZoiaA. Rosso and K. Kardar, Fractional Laplacian in bounded domains, Phys. Rev. E, 76 (2007), 021116.  doi: 10.1103/PhysRevE.76.021116.

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