Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 90, pp. 1-32.

Title: Existence of solutions for non-local elliptic systems with
       Hardy-Littlewood-Sobolev critical nonlinearities

Authors: Yang Yang.   (Jiangnan Univ., Wuxi, Jiangsu, China)
         Qian Yu Hong (Jiangnan Univ., Wuxi, Jiangsu, China)
         Xudong Shang (Nanjing Normal Univ., Taizgou, Jiangsu, China)
 
Abstract:
 In this work, we establish the existence of solutions for the  nonlinear
 nonlocal system of equations involving the fractional Laplacian,
 \begin{gather*}
 \begin{aligned}
 (-\Delta)^s u
 & = au+bv+\frac{2p}{p+q}\int_{\Omega}\frac{|v(y)|^q}{|x-y|^\mu}dy|u|^{p-2}u \\
 &\quad +2\xi_1\int_{\Omega}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}dy|u|^{2^*_\mu-2}u\quad
 \text{in } \Omega,\\
 (-\Delta)^s v
 & = bu+cv+\frac{2q}{p+q}\int_{\Omega}\frac{|u(y)|^p}{|x-y|^\mu}dy|v|^{q-2}v \\
 &\quad +2\xi_2\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}dy|v|^{2^*_\mu-2}v\quad
 \text{in } \Omega,
 \end{aligned}\\
 u =v=0 \quad\text{in }   \mathbb{R}^N\setminus\Omega,
 \end{gather*}
 where $(-\Delta)^s$ is the  fractional Laplacian operator,
 $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $0<s<1$, $N>2s$, 
 $0<\mu<N$,  $\xi_1,\xi_2\geq 0$, $1<p,q\leq 2^*_\mu$ and 
 $2^*_\mu=\frac{2N-\mu}{N-2s}$  is the upper critical exponent in 
 the Hardy-Littlewood-Sobolev inequality.
 The nonlinearities can interact with the spectrum of the fractional Laplacian.
 More specifically, the interval defined by the two eigenvalues of the real
 matrix from the linear part contains an eigenvalue of the spectrum of
 the fractional Laplacian. In this case, resonance phenomena can occur.

Submitted December 11, 2018. Published July 19, 2019.
Math Subject Classifications: 35R11, 35R09, 35A15.
Key Words: Fractional Laplacian; Choquard equation; Linking theorem;
           Hardy-Littlewood-Sobolev critical exponent; Mountain Pass theorem.