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In this paper we study the existence of multiple nontrivial positive weaksolutions to the following system of problems. \begin{align*} \begin{split}-\Delta_{p}u-\Delta_q u &= \lambdaf(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 (C):~$0<\alpha<1,\;0<\beta<1,$$2-\alpha-\beta<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. Wewill guarantee the existence of a solution in the Nehari manifold. Further byusing the Lusternik-Schnirelman category we will prove the existence of atleast $\text{cat}(\Omega)+1$ number of solutions.

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