Sufficient conditions for unique global solutions in optimal control of semilinear equations with $C^1-$nonlinearity

We consider a $C^1-$semilinear elliptic optimal control problem possibly subject to control and/or state constraints. Generalizing previous work we provide a condition which guarantees that a solution of the necessary first order conditions is a global minimum. A similiar result also holds at the discrete level where the corresponding condition can be evaluated explicitly. Our investigations are motivated by G"unter Leugering, who raised the question whether our previous results can be extended to the nonlinearity $\varphi (s)=s\mathrm{\mid }s\mathrm{\mid }$. We develop a corresponding analysis and present several numerical test examples demonstrating its usefulness in practice.