# American Institute of Mathematical Sciences

November  2017, 37(11): 5731-5746. doi: 10.3934/dcds.2017248

## A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions

 1 Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38200 -La Laguna, Spain 2 CONICET-Departamento de Matemática y Estadística, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350, (C1428BCW), C. A. de Buenos Aires, Argentina 3 Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica, Molecular y Fotónica, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38200 -La Laguna, Spain 4 Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avda. España, 1680 -Valparaíso, Chile

* Corresponding author

Received  May 2017 Published  July 2017

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation
 \begin{align} (-\Delta)^s u= h(x_N)f(u) \hbox{in }\mathbb{R}^{N}\end{align}
where
 $(-\Delta)^s$
stands for the fractional Laplacian with
 $s∈ (0, 1)$
, and the functions
 $h$
and
 $f$
are nondecreasing. The main feature is that the function
 $h$
 $\mathbb R$
, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
Citation: Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248
##### References:
 [1] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Var. Partial Differential Equations, 1 (1993), 439-475.  doi: 10.1007/BF01206962. [2] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.  doi: 10.1006/jfan.1996.0125. [3] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.  doi: 10.1006/jdeq.1998.3440. [4] B. Barrios, L. Del Pezzo, J. García-Melián and A. Quaas, Monotonicity of solutions for some nonlocal elliptic problems in half-spaces, Calc. Var. Partial Differential Equations, 56 (2017), 16pp. doi: 10.1007/s00526-017-1133-9. [5] B. Barrios, L. Del Pezzo, J. García-Melián and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, To appear in Rev. Mat. Iberoamericana Available from: https://arxiv.org/abs/1506.04289 [6] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023. [7] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 553-572.  doi: 10.1007/BF01210623. [8] G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbb R^n$ or $\mathbb R^n_+$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315. [9] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9. [10] L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226. [11] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [12] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integrodifferential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274. [13] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8. [14] W. X. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math.(2), 145 (1997), 547-564.  doi: 10.2307/2951844. [15] W. X. Chen and C. Li, Moving planes, moving spheres, and a priori estimates, J. Differential Equations, 195 (2003), 1-13.  doi: 10.1016/j.jde.2003.06.004. [16] W. Chen, C. Li and Y. Li, A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math., 27 (2016), 1650064 (20 pages). doi: 10.1142/S0129167X16500646. [17] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math, 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [18] W. Chen, Y. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022. [19] W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029. [20] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [21] L. M. Del Pezzo and A. Quaas, Global bifurcation for fractional p-Laplacian and an application, Z. Anal. Anwend., 35 (2016), 411-447.  doi: 10.4171/ZAA/1572. [22] J. Dou and H. Zhou, Liouville theorems for fractional Hénon equation and system on $\mathbb R^n$, Comm. Pure Appl. Anal., 14 (2015), 1915-1927.  doi: 10.3934/cpaa.2015.14.1915. [23] Y. Du and S. Li, Nonlinear Liouville Theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations, 10 (2005), 841-860. [24] M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space, Comm. Contemp. Math. , 18 (2016), 1550012 (25 pages). doi: 10.1142/S0219199715500121. [25] M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 1650005 (24 pages). doi: 10.1142/S021919971650005X. [26] 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. [27] Y. Fu and B. Li, On fractional Laplacian problems with indefinite nonlinearity, Applicable Analysis, (2016), 1-17.  doi: 10.1080/00036811.2016.1249861. [28] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406. [29] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196. [30] S. Goyal and K. Sreenadh, Existence of multiple solutions of $p$-fractional Laplace operator with sign-changing weight function, Adv. Nonlinear Anal., 4 (2015), 37-58.  doi: 10.1515/anona-2014-0017. [31] Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551. [32] C. S. Lin, On Liouville theorem and a priori estimates for the scalar curvature equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4), 27 (1998), 107-130. [33] A. Quaas, A. Salort and A. Xia, Principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term, preprint. Available from: http://arxiv.org/abs/1605.09787 [34] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [35] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 257-302.  doi: 10.1016/j.matpur.2013.06.003. [36] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [37] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, preprint, Available from: http://arxiv.org/abs/1401.7402

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##### References:
 [1] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Var. Partial Differential Equations, 1 (1993), 439-475.  doi: 10.1007/BF01206962. [2] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.  doi: 10.1006/jfan.1996.0125. [3] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.  doi: 10.1006/jdeq.1998.3440. [4] B. Barrios, L. Del Pezzo, J. García-Melián and A. Quaas, Monotonicity of solutions for some nonlocal elliptic problems in half-spaces, Calc. Var. Partial Differential Equations, 56 (2017), 16pp. doi: 10.1007/s00526-017-1133-9. [5] B. Barrios, L. Del Pezzo, J. García-Melián and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, To appear in Rev. Mat. Iberoamericana Available from: https://arxiv.org/abs/1506.04289 [6] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023. [7] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 553-572.  doi: 10.1007/BF01210623. [8] G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbb R^n$ or $\mathbb R^n_+$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315. [9] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9. [10] L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226. [11] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [12] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integrodifferential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274. [13] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8. [14] W. X. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math.(2), 145 (1997), 547-564.  doi: 10.2307/2951844. [15] W. X. Chen and C. Li, Moving planes, moving spheres, and a priori estimates, J. Differential Equations, 195 (2003), 1-13.  doi: 10.1016/j.jde.2003.06.004. [16] W. Chen, C. Li and Y. Li, A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math., 27 (2016), 1650064 (20 pages). doi: 10.1142/S0129167X16500646. [17] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math, 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [18] W. Chen, Y. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022. [19] W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029. [20] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [21] L. M. Del Pezzo and A. Quaas, Global bifurcation for fractional p-Laplacian and an application, Z. Anal. Anwend., 35 (2016), 411-447.  doi: 10.4171/ZAA/1572. [22] J. Dou and H. Zhou, Liouville theorems for fractional Hénon equation and system on $\mathbb R^n$, Comm. Pure Appl. Anal., 14 (2015), 1915-1927.  doi: 10.3934/cpaa.2015.14.1915. [23] Y. Du and S. Li, Nonlinear Liouville Theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations, 10 (2005), 841-860. [24] M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space, Comm. Contemp. Math. , 18 (2016), 1550012 (25 pages). doi: 10.1142/S0219199715500121. [25] M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 1650005 (24 pages). doi: 10.1142/S021919971650005X. [26] 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. [27] Y. Fu and B. Li, On fractional Laplacian problems with indefinite nonlinearity, Applicable Analysis, (2016), 1-17.  doi: 10.1080/00036811.2016.1249861. [28] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406. [29] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196. [30] S. Goyal and K. Sreenadh, Existence of multiple solutions of $p$-fractional Laplace operator with sign-changing weight function, Adv. Nonlinear Anal., 4 (2015), 37-58.  doi: 10.1515/anona-2014-0017. [31] Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551. [32] C. S. Lin, On Liouville theorem and a priori estimates for the scalar curvature equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4), 27 (1998), 107-130. [33] A. Quaas, A. Salort and A. Xia, Principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term, preprint. Available from: http://arxiv.org/abs/1605.09787 [34] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [35] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 257-302.  doi: 10.1016/j.matpur.2013.06.003. [36] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [37] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, preprint, Available from: http://arxiv.org/abs/1401.7402
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