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

## TL;DR

This paper proves a Liouville theorem for positive solutions of indefinite fractional diffusion equations with sign-changing coefficients, and applies it to establish existence results for boundary value problems.

## Contribution

It introduces a Liouville theorem for indefinite fractional equations with sign-changing coefficients, enabling new existence results for related boundary value problems.

## Key findings

- Liouville theorem established for indefinite fractional equations
- A priori bounds for positive solutions derived
- Existence of solutions demonstrated via bifurcation methods

## Abstract

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-\Delta)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-\Delta)^s$ stands for the fractional Laplacian with $s\in (0,1)$, 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.