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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Positive solutions of systems of semilinear elliptic equations: the pendulum method
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by Joseph Glover PDF
Trans. Amer. Math. Soc. 301 (1987), 327-342 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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