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
$h$
changes sign in
$\mathbb R$
, 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. BerestyckiI. 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. BerestyckiI. 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. CaffarelliJ. 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. ChenC. 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. ChenY. 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

show all references

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. BerestyckiI. 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. BerestyckiI. 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. CaffarelliJ. 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. ChenC. 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. ChenY. 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

Figure 1.  A possible configuration for $\Omega^+$ and $\Omega^-$ in hypotheses (A).
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