  In this paper we study a system of boundary value problems involving weak
p-Laplacian on the Sierpi\'nski gasket in $\mathbb{R}^2$. Parameters $\lambda,
\gamma, \alpha, \beta$ are real and $1<q<p<\alpha+\beta.$ Functions $a,b,h :
\mathcal{S} \rightarrow \mathbb{R}$ are suitably chosen. For $p>1$ we show the
existence of at least two nontrivial weak solutions to the system of equations
for some $(\lambda,\gamma) \in \mathbb{R}^2.$
