Existence and multiplicity of solutions for superquadratic noncooperative variational elliptic systems (Q1284660)

There are established some existence results for nontrivial solutions of the noncooperative elliptic system \[ -\Delta u = \alpha u - \delta v + F_u(u,v)\text{ in } \Omega, \quad \Delta v = -\delta u - \gamma v + F_v(u,v)\text{ in }\Omega, \quad u=v=0\text{ on }\partial\Omega, \tag{ES} \] where \(\Omega\) is a bounded open domain in \(\mathbb{R}^N\) with smooth boundary, \(\alpha, \delta, \gamma\) are positive parameters and \(F\in C^1(\mathbb{R}^2, \mathbb{R})\). The solutions of (ES) represent the steady state solutions of reaction-diffusion systems. In particular, (ES) can become a \(\lambda-\omega\) system or a FitzHugh-Nagumo system. Furthermore, the multiplicity of solutions for the system (ES) is studied.