Optimal control of nonlinear elliptic problems with sparsity

We study the minimization of the cost functional $$F(\mu )=\parallel u-u_d{\parallel }_{L^p(\mathrm{\Omega })}+\alpha \parallel \mu {\parallel }_{\mathcal{M}(\mathrm{\Omega })},$$ where the controls $\mu$ are taken in the space of finite Borel measures and $u \in W_0^{1, 1}(\Omega)$satisfiestheequation$- \Delta u + g(u) = \mu$ in the sense of distributions in $\mathrm{\Omega }$ for a given nondecreasing continuous function $g:\mathbb{R}\to \mathbb{R}$ such that $g(0)=0$. We prove that $F$ has a minimizer for every desired state $u_d\in L^1(\mathrm{\Omega })$ and every control parameter $\alpha >0$. We then show that when $u_d$ is nonnegative or bounded, every minimizer of $F$ has the same property.