Electron. J. Differential Equations, Vol. 2019 (2019), No. 90, pp. 1-32.

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

Yang Yang, Qian Yu Hong, Xudong Shang

Abstract:
In this work, we establish the existence of solutions for the nonlinear nonlocal system of equations involving the fractional Laplacian,
$$ \displaylines{ (-\Delta)^s u = au+bv+\frac{2p}{p+q}\int_{\Omega}\frac{|v(y)|^q}{|x-y|^\mu}dy|u|^{p-2}u +2\xi_1\int_{\Omega}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}dy|u|^{2^*_\mu-2}u\quad \text{in } \Omega,\cr (-\Delta)^s v = bu+cv+\frac{2q}{p+q}\int_{\Omega}\frac{|u(y)|^p}{|x-y|^\mu}dy|v|^{q-2}v +2\xi_2\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}dy|v|^{2^*_\mu-2}v\quad \text{in } \Omega, \cr u =v=0 \quad\text{in } \mathbb{R}^N\setminus\Omega, }$$
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.

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Yang Yang
School of Science
Jiangnan University
Wuxi, Jiangsu 214122, China
email: yynjnu@126.com
Qian Yu Hong
School of Science
Jiangnan University
Wuxi, Jiangsu 214122, China
email: 1031369190@qq.com
Xudong Shang
School of Mathematics
Nanjing Normal University, Taizhou College
Taizgou, Jiangsu 225300, China
email: xudong-shang@163.com

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