Summary: We consider the optimization problem \[ \max\{{\mathcal E}(\mu):\mu\text{ nonnegative measure},\;\int d\mu=m\}, \] where \({\mathcal E}(\mu)\) is the energy associated to \(\mu\): \[ {\mathcal E}(\mu)= \inf\Biggl\{{1\over 2} \int|Du|^2 d\mu-\langle f,u\rangle: u\in{\mathcal D}(\mathbb{R}^n)\Biggr\}. \] The datum \(f\) is a signed measure with finite total variation and zero average. We show that the optimization problem above admits a solution which is not in \(L^1(\mathbb{R}^n)\) in general. This solution comes out by solving a suitable Monge-Kantorovich equation.