Some optimization problems for nonlinear elastic membranes

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional $\J(u)=\int_{\partial\Omega} f(x) u \rd \H^{N-1}$ over some admissible class of loads $f$ where $u$ is the (unique) solution to the problem $-\Delta_p u + |u|^{p-2}u = 0$ in $\Omega$ with $|\nabla u|^{p-2}u_\nu = f$ on $\partial \Omega$.