Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

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.