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

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} (-Δ)^α u+u^p=0\ \ \ \ &\ {\rm in}\ \ B_1(e_N),\\[2mm]\phantom{(-Δ)^α +u^p} u=δ_{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. $$ \arraycolsep=1pt \begin{array}{lll} -Δu+ u^p=0\ \ \ \ &\ {\rm in}\ \B_1(e_N),\\[2mm] \phantom{-Δ+u^{p} } u=δ_{0}& \ {\rm in}\ \ \R^N\setminus B_1(e_N), \end{array} $$ for which it has been proved that there are no solutions for $p\ge \frac{N+1}{N-1}$.

page 20

References in corpus (1)

• Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary