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