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} -Δ_{p}u-Δ_q u &= λf(x)|u|^{r-2}u+ν\frac{1-α}{2-α-β}h(x) |u|^{-α}|v|^{1-β}\,\,\mbox{in}\,\,Ω,\\ -Δ_{p}v-Δ_q v &= μg(x)|v|^{r-2}v+ν\frac{1-β}{2-α-β}h(x) |u|^{1-α}|v|^{-β}\,\,\mbox{in}\,\,Ω,\\ u,v&>0\,\,\mbox{in}\,\,Ω,\\ u= v &= 0\,\, \mbox{on}\,\, \partialΩ\end{split} \end{align*} where (C):~$0<α<1,\;0<β<1,$ $2-α-β<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. 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 $\text{cat}(Ω)+1$ number of solutions.

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