On the optimization of the first weighted eigenvalue

For N >= 2, a bounded smooth domain Omega in R-N, and g(0), V-0 is an element of L-loc(1)(Omega), we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: -Delta(p)phi + V vertical bar phi vertical bar(p-2)phi = lambda g vertical bar phi vertical bar(p-2)phi in Omega, phi = 0 on partial derivative Omega, where g and V vary over the rearrangement classes of g(0) and V-0, respectively. We prove the existence of a minimizing pair (g, V) and a maximizing pair ((g) over bar, (V) over bar) for g(0) and V-0 lying in certain Lebesgue spaces. We obtain various qualitative properties such as polarization invariance, Steiner symmetry of the minimizers as well as the associated eigenfunctions for the case p = 2. For annular domains, we prove that the minimizers and the corresponding eigenfunctions possess the foliated Schwarz symmetry.