Optimization for the first weighted eigenvalue of local-nonlocal operators with a potential

In this work, we study the optimization problem for the first eigenvalue of mixed local nonlocal operators plus a potential $V$. We begin by investigating the existence and fundamental properties of the eigenvalues, with special emphasis on the first eigenvalue. Finally, we discuss the dependence of the first eigenvalue on the potential function and establish the existence of optimal potentials within certain admissible classes. In particular, we show the existence of a unique maximizer and a minimizer of the first eigenvalue on any bounded, closed, and convex subset of $L^q(\Omega)$. Moreover, these results enable us to characterize the maximizers and minimizers in the closed unit ball of $L^q(\Omega)$, as well as in the class of rearrangements of any $V\in L^q(\Omega)$.