Let Q : ℓ² → ℓ² be a symmetric and positive semi-definite linear operator and fj : R → R (j = 1, 2, ...) be real functions so that, fj(0) = 0 and, for every x = (x1, x2, ....) ∈ ℓ², it holds that f(x) := (f1(x1), f2(x2), ...) ∈ ℓ². Sufficient conditions for the existence of non-trivial solutions to the semilinear problem Qx = f(x) are provided. Moreover, if G is a group of orthogonal linear automorphisms of ℓ² which commute with Q, then such sufficient conditions ensure the existence of non-trivial solutions which are invariant under G. As a consequence, sufficient conditions to ensure solutions of nonlinear partial difference equations on finite degree graphs with vertex set being either finite or infinitely countable are obtained. We consider adaptations to graphs of both Matukuma type equations and Helmholtz equations and study the existence of their solutions.