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

Begoña Barrios; Leandro Del Pezzo; Jorge García-Melián; Alexander Quaas

doi:10.3934/dcds.2017248

Article Contents

2017, Volume 37, Issue 11: 5731-5746. Doi: 10.3934/dcds.2017248

This issue Previous Article On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones Next Article Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

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
* Corresponding author

Received: May 2017

Published: October 2017

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 45M20, 47G10; Secondary: 35B45.

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	S. Alama and G. Tarantello , On semilinear elliptic equations with indefinite nonlinearities, Var. Partial Differential Equations, 1 (1993) , 439-475. doi: 10.1007/BF01206962.
	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.
	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.
	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.
	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
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	W. X. Chen and C. Li , A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997) , 547-564. doi: 10.2307/2951844.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	Y. Du and S. Li , Nonlinear Liouville Theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations, 10 (2005) , 841-860.
	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.
	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.
	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.
	Y. Fu and B. Li , On fractional Laplacian problems with indefinite nonlinearity, Applicable Analysis, (2016) , 1-17. doi: 10.1080/00036811.2016.1249861.
	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.
	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.
	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.
	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.
	C. S. Lin , On Liouville theorem and a priori estimates for the scalar curvature equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1998) , 107-130.
	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
	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.
	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.
	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.
	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