Positive solutions of systems of semilinear elliptic equations: the pendulum method

Conditions are formulated which guarantee the existence of positive solutions for systems of the form

, where $\Delta$ is the Laplacian (with Dirichlet boundary conditions) on an open domain in ${\mathbf{R}^d}$, and where each ${\mu_i}$ is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where $\partial {f_k}/\partial {x_j} > 0$ for every $k$ and $j$.