A bilinear pointwise tracking optimal control problem for a semilinear elliptic PDE

We consider a bilinear optimal control problem with pointwise tracking for a semilinear elliptic PDE in two and three dimensions. The control variable enters the PDE as a (reaction) coefficient and the cost functional contains point evaluations of the state variable. These point evaluations lead to an adjoint problem with a linear combination of Dirac measures as a forcing term. In Lipschitz domains, we derive the existence of optimal solutions and analyze first and necessary and sufficient second order optimality conditions. We also prove that every locally optimal control $\bar u$ belongs to $H^1(\Omega)$. Finally, assuming that the domain $\Omega \subset \mathbb{R}^2$ is a convex polygon, we prove that $\bar u \in C^{0,1}(\bar \Omega)$.