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

## Abstract:

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.
References
• Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
• Vieri Benci and Paul H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. MR 537061, DOI 10.1007/BF01389883
• D. G. Costa and C. A. Magalhães. A variational approach to subquadratic perturbations of elliptic systems, Preprint.
• Ph. Clément, D. G. de Figueiredo, and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), no. 5-6, 923–940. MR 1177298, DOI 10.1080/03605309208820869
• —, Estimates of positive solutions of systems via Hardy-Sobolev inequalities, Preprint.
• Patricio L. Felmer, Periodic solutions of “superquadratic” Hamiltonian systems, J. Differential Equations 102 (1993), no. 1, 188–207. MR 1209982, DOI 10.1006/jdeq.1993.1027
• Daisuke Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 82–86. MR 216336
• B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. MR 619749, DOI 10.1080/03605308108820196
• David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
• J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I, Dunod, Paris, 1968.
• Arne Persson, Compact linear mappings between interpolation spaces, Ark. Mat. 5 (1964), 215–219 (1964). MR 166598, DOI 10.1007/BF02591123
• Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785, DOI 10.1090/cbms/065
• M. A. Souto, Ph.D. Thesis, UNICAMP, Brasil, 1992.
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