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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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On superquadratic elliptic systems
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by Djairo G. de Figueiredo and Patricio L. Felmer PDF
Trans. Amer. Math. Soc. 343 (1994), 99-116 Request permission


In this article we study the existence of solutions for the elliptic system \[ \begin {array}{*{20}{c}} { - \Delta u = \frac {{\partial H}}{{\partial v}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ { - \Delta v = \frac {{\partial H}}{{\partial u}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ {u = 0,\quad v = 0\quad {\text {on}}\;\partial \Omega .} \\ \end {array} \] where $\Omega$ is a bounded open subset of ${\mathbb {R}^N}$ with smooth boundary $\partial \Omega$, and the function $H:{\mathbb {R}^2} \times \bar \Omega \to \mathbb {R}$, is of class ${C^1}$. We assume the function H has a superquadratic behavior that includes a Hamiltonian of the form \[ H(u,v) = |u{|^\alpha } + |v{|^\beta }\quad {\text {where}}\;1 - \frac {2}{N} < \frac {1}{\alpha } + \frac {1}{\beta } < 1\;{\text {with}}\;\alpha > 1,\beta > 1.\] We obtain existence of nontrivial solutions using a variational approach through a version of the Generalized Mountain Pass Theorem. Existence of positive solutions is also discussed.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 343 (1994), 99-116
  • MSC: Primary 35J50; Secondary 35J55, 35J65, 58E05
  • DOI:
  • MathSciNet review: 1214781