Existence and multiplicity of solutions to a $p-q$ Laplacian system with a concave and singular nonlinearities

In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -\Delta_{p}u-\Delta_q u &= \lambda f(x)|u|^{r-2}u+\nu\frac{1-\alpha}{2-\alpha-\beta}h(x) |u|^{-\alpha}|v|^{1-\beta},,\mbox{in},,\Omega,\ -\Delta_{p}v-\Delta_q v &= \mu g(x)|v|^{r-2}v+\nu\frac{1-\beta}{2-\alpha-\beta}h(x) |u|^{1-\alpha}|v|^{-\beta},,\mbox{in},,\Omega,\ u,v&>0,,\mbox{in},,\Omega,\ u= v &= 0,, \mbox{on},, \partial\Omega \end{split} \end{align*} where ©:~$0<\alpha <1,0<\beta <1,$ $2-\alpha -\beta <q<\frac{N(p-1)}{N-p}<p<r<p^{\ast }$, with $p^{\ast }=\frac{Np}{N-p}$. We will guarantee the existence of a solution in the Nehari manifold. Further by using the Lusternik-Schnirelman category we will prove the existence of at least $\text{cat}(\mathrm{\Omega })+1$ number of solutions.