### On positive viscosity solutions of fractional Lane-Emden systems

DOI: http://dx.doi.org/10.12775/TMNA.2019.005

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#### References

N. Abatangelo, Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst. 35 (2015), 5555–5607.

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349–381.

D. Applebaum, textitLévy processes – from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), 1336–1347.

B. Barrios, E. Colorado, A. de Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), 6133–6162.

M. Birkner, J.A. Lópes-Mimbela and A. Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 83–97.

M. Bonforte and V. J. Luis, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal. 218 (2015), 317–362.

C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh 143 (2013), 39–71.

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I Regularity, maximum principles, and Hamiltonian estimates, . Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.

W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal. 120 (2015), 127–153.

W. Choi and S. Kim, Minimal energy solutions to the fractional Lane-Enden system, I. Existence and singularity formation, arXiv: 1610.02853, 2016.

Ph. Clément, D.G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), 923–940.

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, (Universitext)-Springer, 2012.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

M.M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal. 263 (2012), 2205–2227.

P. Felmer and S. Martı́nez, Existence and uniqueness of positive solutions to certain differential systems, Adv. Differential Equations 4 (1998), 575–593.

P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal. 78 (2012), 123–144.

M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z. 279 (2015), 779–809.

D.G. de Figueiredo, Semilinear elliptic systems, Nonlinear Funct. Anal. Appl. Diferential Equations, World Sci. Publishing, River Edge (1998), 122–152.

D.G. de Figueiredo and P. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994), 99–116.

D.G. Figueiredo and B. Ruf, Elliptic systems with nonlinearities of arbitrary growth, Mediterr. J. Math. 1 (2004), 417–431.

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883–901.

J. Hulshof and R. van der Vorst, Diferential Systems with Strongly Indefinite Variational Structure, J. Funct. Anal. 114 (1993), 32–58.

T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist. 22 (2002), 419–441.

E. Leite and M. Montenegro, A priori bounds and positive solutions for non-variational fractional elliptic systems, Differential Integral Equations 30 (2017), 947–974.

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), 125–151.

M. Montenegro, The construction of principal spectra curves for Lane-Emden systems and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (2000), 193–229.

F. Petitta, Some remarks on the duality method for integro-differential equations with measure data, Adv. Nonlinear Stud. 16 (2016), 115–124.

S.I. Pohozaev, On the eigenfunctions of the equation ∆u + λf (u) = 0, Dokl. Akad. Nauk SSSR 165 (1965), 1408–1411.

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 526 (2014), 1–19.

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.

X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations 50 (2014), 723–750.

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), 587–628.

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), 2105–2137.

R. Servadei and E. Valdinoci, A Brezis–Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal. 12 (2013), 2445–2464.

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 (2014), 133–154.

R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102.

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math. 60 (2007), 67–112.

E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, New York, 1970.

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