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Transactions of the American Mathematical Society

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On superquadratic elliptic systems


Authors: Djairo G. de Figueiredo and Patricio L. Felmer
Journal: Trans. Amer. Math. Soc. 343 (1994), 99-116
MSC: Primary 35J50; Secondary 35J55, 35J65, 58E05
MathSciNet review: 1214781
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Abstract: In this article we study the existence of solutions for the elliptic system

\begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u = \frac{{\partial H}}{{\... ...quad v = 0\quad {\text{on}}\;\partial \Omega .} \\ \end{array} \end{displaymath}

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

$\displaystyle H(u,v) = \vert u{\vert^\alpha } + \vert v{\vert^\beta }\quad {\te... ... \frac{1}{\alpha } + \frac{1}{\beta } < 1\;{\text{with}}\;\alpha > 1,\beta > 1.$

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

DOI: https://doi.org/10.1090/S0002-9947-1994-1214781-2
Keywords: Elliptic systems, positive solutions, variational method
Article copyright: © Copyright 1994 American Mathematical Society