# American Institute of Mathematical Sciences

March  2019, 39(3): 1517-1531. doi: 10.3934/dcds.2019065

## A priori bounds and existence result of positive solutions for fractional Laplacian systems

 School of Applied Mathematics, Xiamen University of Technology, 600 Ligong Road, Xiamen 361024, China

* Corresponding author: Lishan Lin

Received  December 2017 Published  December 2018

Fund Project: The author is supported by High-level Talent Project of Xiamen University of Technology grant YKJ14028R.

In this paper, we consider the fractional Laplacian system
 $\left\{\begin{array}{ll}(-\triangle)^{\frac{\alpha}{2}}u+\sum^{N}_{i = 1}b_{i}(x)\frac{\partial u}{\partial x_{i}}+C(x)u = f(x,v), \;\;x\in \Omega,\\(-\triangle)^{\frac{\beta}{2}}v+\sum^{N}_{i = 1}c_{i}(x)\frac{\partial v}{\partial x_{i}}+D(x)v = g(x,u),\;\; x\in \Omega,\\u>0, v>0, \;\; x\in \Omega,\\u = 0, v = 0, \;\; x\in \mathbb R^{N}\setminus \Omega,\end{array}\right.$
where
 $Ω$
is a smooth bounded domain in
 $\mathbb R^{N}$
,
 $α ∈ (1,2)$
,
 $β ∈ (1,2)$
,
 $N>\max\{α, β\}$
. Under some suitable conditions on potential functions and nonlinear terms, we use scaling method to obtain a priori bounds of positive solutions for the fractional Laplacian system with distinct fractional Laplacians.
Citation: Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065
##### References:
 [1] W. Ao, J. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst. Series A, 37 (2017), 5561-5601.  doi: 10.3934/dcds.2017242. [2] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. [3] B. Barrios, I. De Bonis, M. Medina and I. Peral, Semilinear problems for the fractional Laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407.  doi: 10.1515/math-2015-0038. [4] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003. [5] B. Barrios, L. Del Pezzo, J. García-Melián and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, arXiv:1506.04289, 2015. [6] J. P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Physics Reports, 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [7] 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. [8] K. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. [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 blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math.(27), 308 (2016), 1650064, 20 pp.  doi: 10.1142/S0129167X16500646. [11] W. Chen, C. Li and P. Ma, The Fractional Laplacian, accepted for publication in World Scientific Publish Company. [12] Y. Chen and J. Su, Resonant problems for fractional Laplacian, Commun. Pure Appl. Anal., 16 (2017), 163-187.  doi: 10.3934/cpaa.2017008. [13] M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp.  doi: 10.1063/1.3701574. [14] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Boca Raton, Chapman Hall/CRC, FL, 2004. [15] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [16] W. Dong, J. Xu and Z. Wei, Infinitely many weak solutions for a fractional Schrödinger equation, Boundary Value Prob., 53 (2014), 14 pp.  doi: 10.1186/s13661-014-0159-6. [17] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schr¨odinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh, Sect. A, 142 (2012), 1237-1262 doi: 10.1017/S0308210511000746. [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. [19] Y. Gou and J. Nie, Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator, Discrete Contin. Dyn. Syst. Series A, 36 (2016), 6873-6898.  doi: 10.3934/dcds.2016099. [20] T. Gou and H. Sun, Solutions of nonlinear Schrödinger equation with fractional Laplacian without the Ambrosetti-Rabinowitz condition, Appl. Math. Comput., 257 (2015), 409-416.  doi: 10.1016/j.amc.2014.09.035. [21] D. Kriventsov, C1, α interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations, 38 (2013), 2081-2106.  doi: 10.1080/03605302.2013.831990. [22] E. Leite and M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, arXiv:1509.01267, 2015. [23] E. Leite and M. Montenegro, A priori bounds and positive solutions for non-variational fractional elliptic systems, Differential Integral Equations, 30 (2017), 947-974. [24] D. Li and Z. Li, A radial symmetry and Liouville theorem for systems involving fractional Laplacian, Front. Math. China, 12 (2017), 389-402.  doi: 10.1007/s11464-016-0517-z. [25] 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. [26] W. Long, S. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations, Discrete Contin. Dyn. Syst. Series A, 36 (2016), 917-939.  doi: 10.3934/dcds.2016.36.917. [27] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8. [28] A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8. [29] A. Quaas and A. Xia, A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Nonlinearity, 29 (2016), 2279-2297.  doi: 10.1088/0951-7715/29/8/2279. [30] A. Quaas and A. Xia, Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian, Commun. Contemp. Math., 20 (2018), 1750032, 22 pp.  doi: 10.1142/S0219199717500328. [31] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032. [32] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105. [33] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4. [34] K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\mathbb{R}^{N}$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86.  doi: 10.1016/j.nonrwa.2014.06.008. [35] Y. Wei and X. Su, On a class of non-local elliptic equations with asymptotically linear term, Discrete Contin. Dyn. Syst., 38 (2018), 6287-6304.  doi: 10.3934/dcds.2018154. [36] L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $α$-harmonic functions in $\Bbb R^n_+$, Discrete Contin. Dyn. Syst., 36 (2016), 1721-1736.  doi: 10.3934/dcds.2016.36.1721.

show all references

##### References:
 [1] W. Ao, J. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst. Series A, 37 (2017), 5561-5601.  doi: 10.3934/dcds.2017242. [2] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. [3] B. Barrios, I. De Bonis, M. Medina and I. Peral, Semilinear problems for the fractional Laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407.  doi: 10.1515/math-2015-0038. [4] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003. [5] B. Barrios, L. Del Pezzo, J. García-Melián and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, arXiv:1506.04289, 2015. [6] J. P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Physics Reports, 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [7] 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. [8] K. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. [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 blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math.(27), 308 (2016), 1650064, 20 pp.  doi: 10.1142/S0129167X16500646. [11] W. Chen, C. Li and P. Ma, The Fractional Laplacian, accepted for publication in World Scientific Publish Company. [12] Y. Chen and J. Su, Resonant problems for fractional Laplacian, Commun. Pure Appl. Anal., 16 (2017), 163-187.  doi: 10.3934/cpaa.2017008. [13] M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp.  doi: 10.1063/1.3701574. [14] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Boca Raton, Chapman Hall/CRC, FL, 2004. [15] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [16] W. Dong, J. Xu and Z. Wei, Infinitely many weak solutions for a fractional Schrödinger equation, Boundary Value Prob., 53 (2014), 14 pp.  doi: 10.1186/s13661-014-0159-6. [17] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schr¨odinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh, Sect. A, 142 (2012), 1237-1262 doi: 10.1017/S0308210511000746. [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. [19] Y. Gou and J. Nie, Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator, Discrete Contin. Dyn. Syst. Series A, 36 (2016), 6873-6898.  doi: 10.3934/dcds.2016099. [20] T. Gou and H. Sun, Solutions of nonlinear Schrödinger equation with fractional Laplacian without the Ambrosetti-Rabinowitz condition, Appl. Math. Comput., 257 (2015), 409-416.  doi: 10.1016/j.amc.2014.09.035. [21] D. Kriventsov, C1, α interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations, 38 (2013), 2081-2106.  doi: 10.1080/03605302.2013.831990. [22] E. Leite and M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, arXiv:1509.01267, 2015. [23] E. Leite and M. Montenegro, A priori bounds and positive solutions for non-variational fractional elliptic systems, Differential Integral Equations, 30 (2017), 947-974. [24] D. Li and Z. Li, A radial symmetry and Liouville theorem for systems involving fractional Laplacian, Front. Math. China, 12 (2017), 389-402.  doi: 10.1007/s11464-016-0517-z. [25] 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. [26] W. Long, S. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations, Discrete Contin. Dyn. Syst. Series A, 36 (2016), 917-939.  doi: 10.3934/dcds.2016.36.917. [27] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8. [28] A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8. [29] A. Quaas and A. Xia, A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Nonlinearity, 29 (2016), 2279-2297.  doi: 10.1088/0951-7715/29/8/2279. [30] A. Quaas and A. Xia, Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian, Commun. Contemp. Math., 20 (2018), 1750032, 22 pp.  doi: 10.1142/S0219199717500328. [31] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032. [32] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105. [33] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4. [34] K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\mathbb{R}^{N}$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86.  doi: 10.1016/j.nonrwa.2014.06.008. [35] Y. Wei and X. Su, On a class of non-local elliptic equations with asymptotically linear term, Discrete Contin. Dyn. Syst., 38 (2018), 6287-6304.  doi: 10.3934/dcds.2018154. [36] L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $α$-harmonic functions in $\Bbb R^n_+$, Discrete Contin. Dyn. Syst., 36 (2016), 1721-1736.  doi: 10.3934/dcds.2016.36.1721.
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