de Figueiredo, Djairo G.; Felmer, Patricio L. On superquadratic elliptic systems. (English) Zbl 0799.35063 Trans. Am. Math. Soc. 343, No. 1, 99-116 (1994). 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. Cited in 176 Documents MSC: 35J50 Variational methods for elliptic systems 35J65 Nonlinear boundary value problems for linear elliptic equations Keywords:existence of nontrivial solutions; generalized mountain pass theorem PDFBibTeX XMLCite \textit{D. G. de Figueiredo} and \textit{P. L. Felmer}, Trans. Am. Math. Soc. 343, No. 1, 99--116 (1994; Zbl 0799.35063) Full Text: DOI