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 $\partial \Omega$.