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

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 $\mathbb{R}^N$, $02s$, $0<\mu<N$, $\xi_1,\xi_2\geq 0$, $1<p,q\leq 2^*_\mu$ 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.

Keywords

• Fractional Laplacian
• Choquard equation
• Linking theorem
• Hardy-Littlewood-Sobolev critical exponent
• Mountain Pass theorem