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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. Alberti, G. 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. Barrios, L. M. Del Pezzo, J. 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. Bartsch, N. 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. Bogdan, T. 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. Caffarelli, J. 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. Chen, C. 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. Chen, C. 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. Poláčik, P. 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. Zhuo, W. Chen, X. 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. Zoia, A. 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. Alberti, G. 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. Barrios, L. M. Del Pezzo, J. 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. Bartsch, N. 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. Bogdan, T. 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. Caffarelli, J. 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. Chen, C. 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. Chen, C. 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. Poláčik, P. 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. Zhuo, W. Chen, X. 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. Zoia, A. 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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