On a class of semilinear fractional elliptic equations involving outside Dirac data

TL;DR

This paper studies weak solutions of a fractional elliptic equation with outside Dirac data, establishing existence, uniqueness, and non-existence results depending on the exponent p, revealing surprising differences from classical elliptic equations.

Contribution

It provides a complete analysis of weak solutions for a fractional elliptic problem with outside Dirac data, including conditions for existence, uniqueness, and asymptotic behavior as the fractional order approaches 1.

Findings

Unique weak solution exists for p > 1 + 2α/N.

No weak solutions for p in [0, 1 + 2α/N].

Solutions vanish as α approaches 1 for p ≥ (N+2)/(N-2).

Abstract

The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^{\alpha} u+u^p=0\ \ \ \ &\ {\rm in}\ \ B_1(e_N),\\[2mm]\phantom{(-\Delta)^{\alpha} +u^p} u=\delta_{0}& \ {\rm in}\ \ \mathbb{R}^N\setminus B_1(e_N), \end{array} \end{equation} where $p>0$, $ (-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ denotes the fractional Laplacian operator in the principle value sense, $B_1(e_N)$ is the unit ball centered at $e_N=(0,\cdots,0,1)$ in $\mathbb{R}^N$ with $N\ge 2$ and $\delta_0$ is the Dirac mass concentrated at the origin. We prove that problem (\ref{00}) admits a unique weak solution when $p> 1+\frac{2\alpha}{N}$. Moreover, if in addition $p\ge \frac{N+2}{N-2}$, the weak solution vanishes as $\alpha\to 1^-$. We also show that problem (\ref{00}) doesn't have any weak solution when $p\in[0, 1+\frac{2\alpha}{N}]$. These results are very surprising since there are in total contradiction with the classical setting, i.e. $$ \arraycolsep=1pt \begin{array}{lll} -\Delta u+ u^p=0\ \ \ \ &\ {\rm in}\ \B_1(e_N),\\[2mm] \phantom{-\Delta +u^{p} } u=\delta_{0}& \ {\rm in}\ \ \R^N\setminus B_1(e_N), \end{array} $$ for which it has been proved that there are no solutions for $p\ge \frac{N+1}{N-1}$.