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

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$(-\mathrm{\Delta }{)}^{s}u=h(x_N)f(u)\text{in }{\mathbb{R}}^N$$ where $(-\mathrm{\Delta }{)}^s$ 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.