We construct classic solutions of the following supercritical nonlinear fractional exterior problem

$ \begin{equation} \label{mainproblem} \left\{ \begin{array}{l} (-\Delta)^{s}u-u^p=0, \ u>0\ \text{in}\ {\mathbb{R}}^N\setminus \overline{B_1}, \\ u=0 \ \ \text{in}\ \ \overline{B_1}, \quad \lim\limits_{|x|\rightarrow\infty}u(x)=0, \end{array} \right. \end{equation} $

(1.1)

where $s\in(0, 1)$, $p>\frac{N+2s}{N-2s}$ and $B_1$ is the unit ball in ${\mathbb{R}}^N$. As usual, the operator $(-\Delta)^s$ is the fractional Laplacian, defined at any point $x\in {\mathbb{R}}^N $ as

$ \begin{equation*} \begin{aligned} (-\Delta)^su(x):&=C(N, s)P.V.\int_{{\mathbb{R}}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy\\ &=C(N, s)\lim\limits_{\varepsilon\rightarrow 0^+}\int_{{\mathbb{R}}^N\setminus B_{\varepsilon}(x)}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy, \end{aligned} \end{equation*} $

here $P.V.$ is a commonly used abbreviation for "in the principal value sense" and $C(N, s)$ is a constant dependent of $N$ and $s$. We refer to [6-7].

For classical Laplacian, namely, $s=1$, which is the Lame-Emden-Fowler equation

$ \begin{equation} \label{mainproblem1} \left\{ \begin{array}{l} \Delta u+u^p=0, \ u>0\ \text{in}\ {\mathbb{R}}^N\setminus \overline{\Omega}, \\ u=0 \ \ \text{on}\ \ \ \partial\Omega, \quad \lim\limits_{|x|\rightarrow\infty}u(x)=0, \end{array} \right. \end{equation} $

(1.2)

where $\Omega$ is a bounded open set with smooth boundary in ${\mathbb{R}}^N$ and $p>1$. Davila etc [4] proved (1.2) has infinitely many solutions with slow decay $O(|x|^{-\frac{2}{p-1}})$ at infinity with either $N\geq 4$ and $p>\frac{N+1}{N-3}$, or $N\geq 3$, $p>\frac{N+2}{N-2}$ and $\Omega$ is symmetric with respect to $N$ coordinate axes. Later, this result was extended to $p>\frac{N+2}{N-2}$ and $\Omega$ is a smooth bounded domain by Davila etc [5]. For fractional Laplacian, we will prove that this result also holds when $s\in (0, 1)$, $p>\frac{N+2s}{N-2s}$ and $B_1$ is the unit ball in ${\mathbb{R}}^N$. For problem (1.1) in general exterior domain, our method not be used to solve it, there exist some obstacles in Remark 1.

Our main results can be stated as follows:

Theorem 1.1 For any $s\in(0, 1)$ and $p>\frac{N+2s}{N-2s}$, there exists a continuum of solutions $u_{\lambda}$, $\lambda>0$, to problem (1.1) such that

$ \begin{equation*} u_{\lambda}(x)=\beta^{\frac{1}{p-1}}|x|^{-\frac{2s}{p-1}}(1+o(1)) \ \ \ \ \text{as}\ \ |x|\rightarrow\infty \end{equation*} $

and $u_{\lambda}(x)\rightarrow 0$ as $\lambda\rightarrow 0$, uniformly in ${\mathbb{R}}^N\setminus \overline{B_1}$.

Theorem 1.2 For any $s\in(0, 1)$, there exists a number $P_{s}>\frac{N+2s}{N-2s}$, such that for any $p\in(\frac{N+2s}{N-2s}, P_{s})$, problem (1.1) has a fast decay solution $u_{p}$, $u_{p}(x)=O(|x|^{2s-N})$ as $|x|\rightarrow+\infty$.

In order to prove Theorem 1.1, we will take $\omega$ as approximation of (1.1) where $\omega$ is a smooth, radially symmetric, entire solution of the following problem

$ \begin{equation}\label{mainterm} (-\Delta)^s\omega -\omega^p=0, \ \omega>0\ \ \text{in }{\mathbb{R}}^N, \ \ \omega(0)=1, \ \ \lim\limits_{|x|\rightarrow\infty}\omega(x)|x|^{\frac{2s}{p-1}}=\beta^{\frac{1}{p-1}}, \end{equation} $

(1.3)

here $\beta$ is a positive constant chosen so that $\beta^{\frac{1}{p-1}}|x|^{-\frac{2s}{p-1}}$ is a singular solution to $(-\Delta)^s\omega -\omega^p=0$ for which the existence and linear theory has been studied recently in [1] for the fractional case.

The basic idea in the proof of Theorem 1.2 is to consider as an initial approximation the function $\lambda^\frac{N-2s}{2}\omega_{**}(\lambda x +\xi), $ where

$ \begin{equation}\label{apoxfunction} \omega_{**}(r)=\left(\frac{1}{1+A_{N, s}r^2}\right)^\frac{N-2s}{2} \end{equation} $

(1.4)

is the unique positive radial smooth solution of the problem

$ \begin{equation*} (-\Delta)^s\omega_{**}=\omega_{**}^\frac{N+2s}{N-2s} \ \ \text{in}\ {\mathbb{R}}^N, \ \ \ \omega_{**}(0)=1. \end{equation*} $

These scalings will constitute good approximations for small $\lambda$ if $p$ is sufficiently close to $\frac{N+2s}{N-2s}$. We prove then adjusting both $\xi$ and $\lambda$, produces a solution as desired after addition of a lower order term.

By the change of variables

$ \begin{equation*} \tilde u(x)=\lambda^{-\frac{2}{p-1}}u\left(\frac{x-\xi}{\lambda}\right) \end{equation*} $

and the maximum principle (see the page 39 of [3]), problem (1.1) is equivalent to

$ \begin{equation} \label{eqproblem} \left\{ \begin{aligned} &(-\Delta)^s\tilde u-|\tilde u|^p=0, \ \tilde u\not\equiv0\hspace{0.5cm} \text{in}\ {\mathbb{R}}^N\setminus \overline{{B_1}_{\lambda, \xi}}, \\ &\tilde u=0 \ \ \text{in}\ \ \overline{{B_1}_{\lambda, \xi}}, \ \lim\limits_{|x|\rightarrow\infty}\tilde u(x)=0 \end{aligned} \right. \end{equation} $

(1.5)

where $\lambda>0$ is a small parameter and $B_{1\lambda, \xi}$ is the shrinking domain

$ \begin{equation*} B_{1\lambda, \xi}=\{\lambda x+\xi\ \big{|}\ x\in B_1\}. \end{equation*} $

Remark 1 To prove Theorem 1.1 and Theorem 1.2, we will construct solutions of the equivalent problem (1.5) with the form $\widetilde{u}=\omega+\varphi_\lambda+\phi$ and $\widetilde{u}=\omega_{**}+\varphi_\lambda+\phi$. To obtain the decay of $\widetilde{u}$, we need to know that the decay of $\varphi_\lambda+\phi$. Using the Poisson Kernel $P(x, y)$ in $R^N\setminus B_1$, we first obtain the decay of $\varphi_\lambda$ is no more than $O(|x-\xi|^{2s-N})$. Secondly, we can derive the decay of $\phi$ by the Green function $G(x, y)$ in $R^N\setminus B_1$. But for general exterior domain, there is a lack of the explicit formulas and the decay of Poisson Kernel and Green's function of fractional Laplace operator$(-\Delta)^s$.

The proof of Theorem 1.1 and Theorem 1.2 refers to [2] in detail.