Nonexistence and uniqueness in semilinear elliptic systems (Q675953)

The author studies systems of semilinear elliptic equations of the form \[ \Delta u_{i}+ f_{i}(x,u_{1},u_{2},\dots,u_{k})= 0 \quad \text{in }\Omega, \qquad u_{i} = 0 \quad \text{on } \partial \Omega \] where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain, \(i=1,2,\dots,k\) and \(f_{1}, f_{2},\dots,f_{k}\) are suitably given functions. Generalizations of the Pokhozaev identity are explored and bounds for the solutions, as well as criteria for nonexistence and uniqueness of solutions are determined.