Certain fractional Laplacian equations that do not have smooth solutions

Abstract

Let $f$ be a real-valued function defined on $\mathbb{R}$, with $f(0)\ne 0$ and which is not constant in non empty open intervals. We prove the equations \begin{equation}\label{edif} \left\{ \begin{array}{rcll} (-Δ)^{s}u & = & f(u), & \text{in }B_{1}, \\ u & = & 0, & \text{in }B_{1}^{c}, \end{array} \right. \end{equation} where $(-\Delta {)}^s$ is the $s$-fractional Laplacian, $0<s<1$, have no solutions in $C^2(\overline{B_1})$, if $d>2s$. The proof is based on the moving plane method and in the approximation of $C^2(\overline{B_1})$ functions by $s$-harmonic functions.

9 pages, 2 figures