  In this manuscript, we will study the asymptotic behavior for a class of
nonlocal diffusion equations associated with the weighted fractional
$\wp(\cdot)-$Laplacian operator involving constant/variable exponent. In the
case of constant exponents, under some appropriate conditions, we will study
the existence of solutions and asymptotic behavior of solutions by employing
the subdifferential approach and we will study the problem when $\wp$ goes to
$\infty$. Already, for case the weighted fractional $\wp(\cdot)$-Laplacian
operator, we will also study the asymptotic behavior of the problem solution
when $\wp(\cdot)$ goes to $\infty$, in the whole or in a subset of the domain
(the problem involving the fractional $\wp(\cdot)$-Laplacian presents a
discontinuous exponent). To obtain the results of the asymptotic behavior in
both problems it will be via Mosco convergence.
