# Certain fractional Laplacian equations that do not have smooth solutions

## TL;DR

This paper proves that certain fractional Laplacian equations with non-constant right-hand functions do not admit smooth solutions within a specific regularity class, using the moving plane method and approximation techniques.

## Contribution

It establishes a non-existence result for smooth solutions of fractional Laplacian equations with non-constant nonlinearities in certain dimensions.

## Key findings

- No solutions in C^2 for d > 2s
- Uses moving plane method for proof
- Employs approximation by s-harmonic functions

## Abstract

Let $f$ be a real-valued function defined on $\mathbb{R}$, with $f(0) \neq 0$ and which is not constant in non empty open intervals. We prove the equations \begin{equation}\label{edif} \left\{ \begin{array}{rcll} (-\Delta )^{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.