Title: Strong comparison principle for the fractional $p$-Laplacian and applications to starshaped rings,
Abstract: In the following we show the strong comparison principle for the fractional
$p$-Laplacian, i.e. we analyze functions $v,w$ which satisfy $v\geq w$ in
$\mathbb{R}^N$ and
\[
(-\Delta)^s_pv+q(x)|v|^{p-2}v\geq (-\Delta)^s_pw+q(x)|w|^{p-2}w \quad
\text{in $D$,}
\] where $s\in(0,1)$, $p>1$, $D\subset \mathbb{R}^N$ is an open set, and
$q\in L^{\infty}(\mathbb{R}^N)$ is a nonnegative function. Under suitable
conditions on $s,p$ and some regularity assumptions on $v,w$ we show that
either $v\equiv w$ in $\mathbb{R}^N$ or $v>w$ in $D$. Moreover, we apply this
result to analyze the geometry of nonnegative solutions in starshaped rings and
in the half space.