  In this paper we prove the existence of an optimal set for the minimization
of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open
sets of fixed measure included in a box of finite measure. An analogous
existence result is obtained for eigenvalues of the $p$-Laplacian associated
with Schr\"odinger potentials. In order to deal with these nonlinear shape
optimization problems, we develop a general approach which allows to treat the
continuous dependence of the eigenvalues of the $p$-Laplacian associated with
sign-changing capacitary measures under $\gamma$-convergence.
