Title: A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions,
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.