Strong comparison principle and symmetry results for the fractional $p$-Laplacian

Abstract

In this article, we study the equation $$(-{\Delta }_p{)}^{s}u=f(u)$$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$, where $n\ge 2$, $p>2$, and $f$ is locally Lipschitz. We establish a strong comparison principle in a fairly general setting and use it to derive symmetry results for positive $C^1$ solutions satisfying Dirichlet boundary conditions. We also show that the $C^1$ regularity assumption is indeed satisfied for $p\in \left[2,\frac{2}{1-s}\right)$.

38 pages, 1 figure