Optimization problems in rearrangement classes for fractional p-Laplacian equations

We discuss two optimization problems related to the fractional p-Laplacian. First, we prove the existence of at least one minimizer for the principal eigenvalue of the fractional p-Laplacian with Dirichlet conditions, with a bounded weight function varying in a rearrangement class. Then, we investigate the minimization of the energy functional for general nonlinear equations driven by the same operator, as the reaction varies in a rearrangement class. In both cases, we provide a pointwise relation between the optimizing datum and the corresponding solution.