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

Abstract

In the paper we consider a boundary value problem involving a differential equation with the fractional Laplacian \begin{document}$ (-Δ)^{α/2}$\end{document} for \begin{document}$ \mathit{\alpha }\in (1,2\rm{)}$\end{document} and some superlinear and subcritical nonlinearity \begin{document}$ G_{z}$\end{document} 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.