In this paper, we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:$$\left\{\begin{array}{lll} (-\Delta)^{\frac{\alpha}{2}}_{a_1} u_1(x)=u_1^{q_{11}}(x)+u_2^{q_{12}}(x)+ h_1(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ (-\Delta)^{\frac{\beta}{2}}_{a_2} u_2(x)=u_1^{q_{21}}(x)+u_2^{q_{22} }(x)+h_2(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ u_1(x)=0,~u_2(x)=0,~~~x\in \mathbb{R}^n \backslash \Omega. \end{array}\right.$$ Here $(-\Delta)^{\frac{\alpha}{2}}_{a_1}$ and $(-\Delta)^{\frac{\beta}{2}}_{a_2}$ denote weighted fractional Laplacians and $\Omega \subset \mathbb{R}^n$ is a $C^2$ bounded domain. It is shown that under some assumptions on $h_i(i=1,2)$, the problem admits at least one positive solution $(u_1(x),u_2(x))$. We first obtain the {a priori} bounds of solutions to the system by using the direct blow-up method of Chen, Li and Li. Then the proof of existence is based on a topological degree theory.
[1] Barrios B, Del Pezzo L, Melián J G, Quaas A. A priori bounds and existence of solutions for some nonlocal elliptic problems. Rev Mat Iberoam, 2018, 34:195-220
[2] Bonforte M, Sire Y, Vázquez J L. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete Contin Dyn Syst, 2015, 35:5725-5767
[3] Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differential Equations, 2007, 32:1245-1260
[4] Caffarelli L, Silvestre L. Regularity theory for fully nonlinear integro-differential equations. Comm Pure Appl Math, 2009, 62:597-638
[5] Caffarelli L, Silvestre L. Regularity results for nonlocal equations by approximation. Arch Ration Mech Anal, 2011, 200:59-88
[6] Caffarelli L, Vasseur A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann Math, 2010, 171:1903-1930
[7] Cao Y, Wu J, Wang L. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete Contin Dyn Syst, 2019, 39:1237-1256
[8] Chang K. Methods in Nonlinear Analysis, Springer Monographs in Mathematics. Berlin:Springer-Verlag, 2005
[9] Chen W, Fang Y, Yang R. Loiuville theorems involving the fractional Laplacian on a half space. Adv Math, 2015, 274:167-198
[10] Chen W, Li C. A priori estimates for solutions to nonlinear elliptic equations. Arch Rational Mech Anal, 1993, 122:145-157
[11] Chen W, Li C, Li Y. A direct blowing-up and rescaling argument on nonlocal elliptic equations. Internat J Math, 2016, 27:1650064, 20 pp
[12] Chen W, Li C, Ou B. Classification of solutions for an integral equation. Comm Pure Appl Math, 2006, 59:330-343
[13] Chen W, Li Y, Ma P. The Fractional Laplacian. World Scientific Publishing Company, 2020
[14] Cheng T, Huang G, Li C. The maximum principles for fractional Laplacian equations and their applications. Commun Contemp Math, 2017, 19:1750018, 12 pp
[15] Constantin P. Euler equations, Navier-Stokes equations and turbulence//Mathematical foundation of turbulent viscous flows. Lecture Notes in Math, 1871. Springer, 2006:1-43
[16] Fall M M. Regularity estimates for nonlocal Schrödinger equations. Discrete Contin Dyn Syst, 2019, 39:1405-1456
[17] de Figueiredo D G, Sirakov B. Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems. Math Ann, 2005, 333:231-260
[18] Gidas B, Spruck J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm Pure Appl Math, 2010, 34:525-598
[19] Gidas B, Spruck J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm Partial Differential Equations, 1981, 6:883-901
[20] Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. 2nd ed. Berlin:SpringerVerlag, 1983
[21] Jarohs S, Weth T. Symmetry via antisymmetric maximum principle in nonlocal problems of variable order. Ann Mat Pura Appl, 2016, 195:273-291
[22] Kriventsov D. C1,α interior regularity for nonlinear nonlocal elliptic equations with rough kernels. Comm Partial Differential Equations, 2013, 38:2081-2106
[23] Li C, Wu Z. Radial symmetry for systems of fractional Laplacian. Acta Math Sci, 2018, 38B:1567-1582
[24] Lin L. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete Contin Dyn Syst, 2019, 39:1517-1531
[25] Leite E, Montenegro M. A priori bounds and positive solutions for nonvariational fractional elliptic systems. Differential Integral Equations, 2017, 30:947-974
[26] Polácik P, Quittner P, Souplet P. Singularity and decay estimates in superlinear problems via Liouville-type theorems, I. Elliptic equations and systems. Duke Math J, 2007, 139:555-579
[27] Pivato M, Seco L. Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis. J Multivariate Anal, 2003, 87:219-240
[28] Quaas A, Xia A. A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian. Nonlinearity, 2016, 29:2279-2297
[29] Quaas A, Xia A. Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space. Calc Var Partial Differential Equations, 2015, 52:641-659
[30] Quaas A, Xia A. Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian. Commun Contemp Math, 2018, 20:1750032, 22 pp
[31] Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60:67-112
[32] Servadei R, Valdinoci E. Weak and viscosity solutions of the fractional Laplace equation. Publ Mat, 2014, 58:133-154
[33] Servadei R, Valdinoci E. Lewy-Stampacchia type estimates for variational inequalities driven by (non) local operators. Rev Mat Iberoam, 2013, 29:1091-1126
[34] Wang P, Wang Y. Positive solutions for a weighted fractional system. Acta Math Sci, 2018, 38B:935-949
[35] Wang P, Niu P. Liouville's Theorem for a fractional elliptic system. Discrete Contin Dyn Syst, 2019, 39:1545-1558
[36] Wang P, Niu P. Symmetric properties for fully nonlinear nonlocal system. Nonlinear Anal, 2019, 187:134-146
[37] Yang J, Yu X. Existence of a cooperative elliptic system involving Pucci operator. Acta Math Sci, 2010, 30B:137-147
[38] Zhuo R, Chen W, Cui X, Yuan Z. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete Contin Dyn Syst, 2016, 36:1125-1141
