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. AoJ. 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. BarriosI. De BonisM. 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. BarriosE. ColoradoR. 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. 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 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 NezzaG. 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. DongJ. 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. LongS. 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. PoláčikP. 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. ZhangC. LiW. 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. AoJ. 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. BarriosI. De BonisM. 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. BarriosE. ColoradoR. 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. 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 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 NezzaG. 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. DongJ. 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. LongS. 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. PoláčikP. 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. ZhangC. LiW. 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.

[1]

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

[2]

Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121

[3]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[4]

Hua Chen, Hong-Ge Chen. Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 301-317. doi: 10.3934/dcds.2021117

[5]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113

[6]

Guowei Dai, Rushun Tian, Zhitao Zhang. Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1905-1927. doi: 10.3934/dcdss.2019125

[7]

Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283

[8]

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

[9]

Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure and Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941

[10]

Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure and Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177

[11]

Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070

[12]

Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193

[13]

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

[14]

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

[15]

Rongrong Yang, Zhongxue Lü. The properties of positive solutions to semilinear equations involving the fractional Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1073-1089. doi: 10.3934/cpaa.2019052

[16]

Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141

[17]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069

[18]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure and Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567

[19]

Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166

[20]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (287)
  • HTML views (104)
  • Cited by (1)

Other articles
by authors

[Back to Top]