  In the first part of this article we deal with the existence of at least
three non-trivial weak solutions of a nonlocal problem with nonstandard growth
involving a nonlocal Robin type boundary condition. The second part of the
article is devoted to study eigenvalues and minimizers of several nonlocal
problems for the fractional $g-$Laplacian $(-\Delta_g)^s$ with different
boundary conditions, namely, Dirichlet, Neumann and Robin.
