Non-variational weakly coupled elliptic systems (Q2119600)

The authors consider a semilinear elliptic system with subcritical nonlinearities on a smooth bounded domain \(\Omega\subset\mathbb{R}^N\), \(N\ge2\). The system is not variational except for special choices of the parameters. The existence of a solution, satisfying homogeneous Dirichlet boundary conditions and where all components are positive, is proved via degree theory, based on a homotopy to the decoupled system (which is still nonlinear). In the case of 2 equations the authors also investigate the existence of synchronized solutions, i.e., solutions \((u_1,u_2) \in H^1_0(\Omega)\times H^1_0(\Omega)\) where \(u_2=tu_1\) for some \(t>0\).