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Transactions of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1994-1214781-2
MathSciNet review: 1214781
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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.


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Keywords: Elliptic systems, positive solutions, variational method
Article copyright: © Copyright 1994 American Mathematical Society