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

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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.

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

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

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