doi: 10.3934/cpaa.2021168
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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 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 & Applied Analysis, doi: 10.3934/cpaa.2021168
References:
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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

show all references

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

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