  In the paper we consider a boundary value problem involving a differential
equation with the fractional Laplacian $(-\Delta)^{\alpha/2}$ for $\alpha
\in\left( 1,2\right) $ and some superlinear and subcritical nonlinearity
$G_{z}$ provided with a nonhomogeneous Dirichlet exterior boundary condition.
Some sufficient conditions under which the set of weak solutions to the
boundary value problem is nonempty and depends continuously in the
Painleve-Kuratowski sense on distributed parameters and exterior boundary data
are stated. The proofs of the existence results rely on the Mountain Pass
Theorem. The application of the continuity results to some optimal control
problem is also provided.
