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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:\[ -\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g |\phi|^{p-2}\phi \text{ in } \Omega, \quad \phi=0 \text{ on } \partial \Omega, \]where $g$
and $V$
vary over the rearrangement classes of $g_0$
and $V_0$
, respectively. We prove the existence of a minimizing pair $(\underline {g},\,\underline {V})$
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.
