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 $u$.