Multiplicity of positive solutions to a singular $(p_1,p_2)$-Laplacian system with coupled integral boundary conditions

In this work, we investigate the existence and multiplicity results for positive solutions to a singular $(p_1,p_2)$-Laplacian system with coupled integral boundary conditions and a parameter $(\mu,\lambda) \in \mathbb{R}_+^3 $. Using sub-super solutions method and fixed point index theorems, it is shown that there exists a continuous surface $\mathcal{C}$ which separates $\mathbb{R}_+^2 \times (0,\infty)$ into two regions $\mathcal{O}_1$ and $\mathcal{O}_2$ such that the problem under consideration has two positive solutions for $( \mu,\lambda) \in \mathcal{O}_1,$ at least one positive solution for $( \mu,\lambda) \in \mathcal{C}$, and no positive solutions for $( \mu,\lambda) \in \mathcal{O}_2.$