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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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 \[ \begin {gathered} - \Delta {u_1} + {f_1}({u_1}, \ldots , {u_n}) = {\mu _1}, \hfill \\ - \Delta {u_2} + {f_2}({u_1}, \ldots , {u_n}) = {\mu _2}, \hfill \\ \vdots \quad \quad \quad \quad \quad \vdots \quad \quad \quad \vdots \quad \vdots \hfill \\ - \Delta {u_n} + {f_n}({u_1}, \ldots , {u_n}) = {\mu _n}, \hfill \\ \end {gathered} \], 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 $j$.


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Article copyright: © Copyright 1987 American Mathematical Society