Let Q be a m  m real matrix and f_j :  → , j = 1, …, m, be some given functions. If x and f(x) are column vectors whose j-coordinates are x_j and f_j(x_j), respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system Qx = f(x) for Q symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by Neuberger. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph G is just any linear operator D:C⁰(G) → C⁰(G), where C⁰(G) is the real vector space of real maps defined on the vertex set V of the graph. Given a derivation D and a function F:V   → , one has associated a partial difference equation Dμ = F(v,μ), and one searches for solutions μ ∈ C⁰(G). Sufficient conditions in order to have non-trivial solutions of partial difference equations on any finite, connected simple graph for D symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair (G, d), where G is a connected finite degree simple graph and d is a positive function on the set of edges of the graph. The metric d permits to consider some classical derivations, such as the Laplacian operator ▵₂. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91-107) was considered the nonlinear elliptic partial difference equations ▵₂u = F(u), for the metric d = 1.