For $N\geq 2$, a bounded smooth domain $\Omega$ in $\mathbb{R}^N$, and $g_0,
V_0 \in L^1_{loc}(\Omega)$, we study the optimization of the first eigenvalue
for the following weighted eigenvalue problem: \begin{align*}
-\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g |\phi|^{p-2}\phi \text{ in }
\Omega, \quad \phi=0 \text{ on } \partial \Omega, \end{align*} where $g$ and
$V$ vary over the rearrangement classes of $g_0$ and $V_0$, respectively. We
prove the existence of a minimizing pair $(\underline{g},\underline{V})$ and a
maximizing pair $(\overline{g},\overline{V})$ 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.
