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On superquadratic elliptic systems. (English) Zbl 0799.35063

Summary: We study the existence of solutions for the elliptic system \[ -\Delta u= {{\partial H} \over {\partial v}} (u,v,x), \qquad -\Delta v= {{\partial H} \over {\partial u}} (u,v,x) \quad \text{in } \Omega, \qquad u=0,\;v=0 \quad \text{on } \partial\Omega, \] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^ N\) with smooth boundary \(\partial\Omega\), and the function \(H: \mathbb{R}^ 2\times \overline{\Omega}\to \mathbb{R}\), is of class \(C^ 1\). We assume the function \(H\) has a superquadratic behavior that includes a Hamiltonian of the form \(H(u,v)= | u|^ \alpha + | v|^ \beta\) where \(1-2/N< 1/\alpha+ 1/\beta<1\) 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.

MSC:

35J50 Variational methods for elliptic systems
35J65 Nonlinear boundary value problems for linear elliptic equations
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