Strongly indefinite functionals and multiple solutions of elliptic systems

We study existence and multiplicity of solutions of the elliptic system

where $\Omega\subset\mathbb{R}^N, N\geq 3$, is a smooth bounded domain and $H\in \mathcal{C}^1(\bar{\Omega}\times\mathbb{R}^2, \mathbb{R})$. We assume that the nonlinear term

where $p\in (1, 2^*)$, $2^*:=2N/(N-2)$, and $q\in (1, \infty)$. So some supercritical systems are included. Nontrivial solutions are obtained. When $H(x,u,v)$ is even in $(u,v)$, we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if $p>2$ (resp. $p<2$). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.