Differential and Integral Equations

Nonexistence of positive solutions for a system of semilinear fractional Laplacian problem

Jingbo Dou and Ye Li

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we consider a system of semilinear equations involving the fractional Laplacian in the Euclidean space $\mathbb{R}^n$: \begin{equation*} \begin{cases} (-\Delta)^{\alpha/2}u(x)=f(x_n)v^p(x)\\ (-\Delta)^{\alpha/2}v(x)=g(x_n)u^q(x) \end{cases} \end{equation*} in the subcritical case $1 < p,q\le \frac{n+\alpha}{n-\alpha}$ where $\alpha \in (0,\,2)$. Instead of investigating the above system directly, we discuss its equivalent integral system: \begin{equation*} \begin{cases} u(x)=\int_{\mathbb{R}^n} G_{\infty}(x,y)f(y_n)v^p(y)dy\\ v(y)=\int_{\mathbb{R}^n} G_{\infty}(x,y)g(x_n)u^q(x)dx , \end{cases} \end{equation*} where $G_{\infty}(x, y)$ is the Green's function associated with the fractional Laplacian in $\mathbb{R}^n$. Under natural structure condition on $f$ and $g$, we indicate the nonexistence of the positive solutions to the above integral system according to the method of moving spheres in integral form and the classic Hardy-Littlewood-Sobolev inequality.

Article information

Differential Integral Equations, Volume 31, Number 9/10 (2018), 715-734.

First available in Project Euclid: 13 June 2018

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B65: Smoothness and regularity of solutions


Dou, Jingbo; Li, Ye. Nonexistence of positive solutions for a system of semilinear fractional Laplacian problem. Differential Integral Equations 31 (2018), no. 9/10, 715--734. https://projecteuclid.org/euclid.die/1528855437

Export citation