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

Glover, Joseph

doi:10.1090/S0002-9947-1987-0879577-6

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

Author: Joseph Glover
Journal: Trans. Amer. Math. Soc. 301 (1987), 327-342
MSC: Primary 35J60; Secondary 35A35
DOI: https://doi.org/10.1090/S0002-9947-1987-0879577-6
MathSciNet review: 879577
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Abstract: Conditions are formulated which guarantee the existence of positive solutions for systems of the form

$\begin{displaymath}\begin{gathered}- \Delta {u_1} + {f_1}({u_1}, \ldots ,\,{u_n}... ...}({u_1}, \ldots ,\,{u_n}) = {\mu _n}, \hfill \\ \end{gathered} \end{displaymath}$

, 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

and

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