Existence and multiplicity of solutions for semilinear elliptic systems involving Hardy-Sobolev critical nonlinearity (Q2868076)

In this paper, the existence of solutions for the problem NEWLINE\[NEWLINE\begin{gathered} -\text{div}(|x|^{-2a}\nabla u) - \mu \frac{u}{|x|^{2(1+a)}} = \frac{2\alpha}{\alpha+\beta} \frac{|u|^{\alpha-2}|v|^\beta u}{|x|^{bp}} + \lambda \frac{\partial}{\partial u} F(x,u,v), \quad x \in \Omega, \\ -\text{div}(|x|^{-2a}\nabla v) - \mu \frac{v}{|x|^{2(1+a)}} = \frac{2\beta}{\alpha+\beta} \frac{|u|^\alpha|v|^{\beta-2} v}{|x|^{bp}} + \lambda \frac{\partial}{\partial v} F(x,u,v), \quad x \in \Omega, \\ u=v=0, \quad x \in \partial \Omega, \end{gathered}NEWLINE\]NEWLINE is established, where \(\Omega\) is a bounded domain in \({\mathbb R}^N\) with smooth boundary, \(N \geq 3\), \(0 \in \Omega\), NEWLINE\[NEWLINE 0 \leq a < \frac{N-2}{2}, \quad 0 \leq \mu < \left( \frac{N-2}{2} -a \right)^2, NEWLINE\]NEWLINE \(a \leq b \leq a+1\), \(\lambda>0\), \(\alpha>1\), \(\beta>1\), and NEWLINE\[NEWLINE \alpha+\beta=\frac{2N}{N-2(1+a-b)} NEWLINE\]NEWLINE is the Hardy-Sobolev critical exponent.