On a \(p\)-Laplacian system with critical Hardy--Sobolev exponents and critical Sobolev exponents (Q1933295)

In this paper, the existence results of positive solutions for the semiliner elliptic system \[ \begin{cases} -\text{div} (|\nabla u_i|^{p-2} \nabla u_i) - \mu \frac{|u_i|^{p-2}u_i}{|x|^p} \\ = \frac{1}{p^*} F_{u_i}(u_1,\ldots,u_k) + \frac{|u_i|^{p^*(t)-2}u_i}{|x|^t} + \lambda \frac{|u_i|^{p-2}u_i}{|x|^s}, \quad x \in \Omega, \\ u_i=0 \quad \text{on} \;\partial \Omega, \\ 1 \leq i \leq k, \end{cases} \] are established, where \(0 \in \Omega\) is a bounded domain in \({\mathbb R}^N\) with the smooth boundary \(\partial \Omega\), \(N \geq 3\), \(1<p<N\), \(0 \leq \mu< ((N-p)/p)^p\), \(\lambda>0\), \(0 \leq t <p\), \(p^*(t)=p(N-t)/(N-p)\) and \(p^*=p^*(0)=pN/(N-p)\).