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

摘要

In the paper we consider a boundary value problem involving a differentialequation 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 theboundary value problem is nonempty and depends continuously in thePainleve-Kuratowski sense on distributed parameters and exterior boundary dataare stated. The proofs of the existence results rely on the Mountain PassTheorem. The application of the continuity results to some optimal controlproblem is also provided.