Optimal control of nonlinear elliptic problems with sparsity

We study the minimization of the cost functional $$F(\mu) = \lVert u - u_d \rVert_{L^p(\Omega)} + \alpha \lVert \mu \rVert_{\mathcal{M}(\Omega)},$$ where the controls $\mu$ are taken in the space of finite Borel measures and $u \in W_0^{1, 1}(\Omega)$ satisfies the equation $- \Delta u + g(u) = \mu$ in the sense of distributions in $\Omega$ 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 $u_d \in L^1(\Omega)$ 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.