Shape optimization problems involving nonlocal and nonlinear operators

In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional $F$ defined on the family of all ‘quasi-open’ subsets of a bounded open set $\mathrm{\Omega }$ in ${\mathbb{R}}^n$. This is ensured under the condition that $F$ demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional $p$-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when $s\to 1$ and extend this result to the anisotropic case.