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.