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

Abstract:

Conditions are formulated which guarantee the existence of positive solutions for systems of the form \[ \begin {gathered} - \Delta {u_1} + {f_1}({u_1}, \ldots , {u_n}) = {\mu _1}, \hfill \\ - \Delta {u_2} + {f_2}({u_1}, \ldots , {u_n}) = {\mu _2}, \hfill \\ \vdots \quad \quad \quad \quad \quad \vdots \quad \quad \quad \vdots \quad \vdots \hfill \\ - \Delta {u_n} + {f_n}({u_1}, \ldots , {u_n}) = {\mu _n}, \hfill \\ \end {gathered} \], 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 $j$.