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 \[ (\Delta)^s_pv+q(x)v^{p2}v\geq (\Delta)^s_pw+q(x)w^{p2}w \quad \text{in $D$,} \] 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 arXiv:
 arXiv:1706.01234
 Bibcode:
 2017arXiv170601234J
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 18 pages, to appear in Advanced Nonlinear Studies