Starshape of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

Abstract

In the present work we study solutions of the problem $-(-\Delta {)}^{\alpha /2}u=f(x,u)$ in $D_0\setminus {\overline{D}}_1$, with exterior conditions $u=0$ in $R^N\setminus D_0$ and $u=1$ in ${\overline{D}}_1$, where $D_1,D_0\subset R^N$ are open sets such that ${\overline{D}}_1\subset D_0$, $\alpha \in (0,2)$, and $f$ is a nonlinearity. Under different assumptions on $f$ we prove that, if $D_0$ and $D_1$ are starshaped with respect to the same point $\bar{x}\in {\overline{D}}_1$, then the same occurs for every superlevel set of $u$.

References in corpus (1)

• On the maximum principle for higher-order fractional Laplacians

Cited by in corpus (1)

• Strong comparison principle for the fractional $p$-Laplacian and applications to starshaped rings