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

TL;DR

This paper investigates boundary value problems involving fractional Laplacians with superlinear nonlinearities, establishing existence and continuous dependence of solutions on parameters using the Mountain Pass Theorem, with applications to optimal control.

Contribution

It applies the Mountain Pass Theorem to fractional Laplacian equations with boundary data, providing new conditions for solution existence and parameter dependence.

Findings

Existence of weak solutions under certain conditions.

Solutions depend continuously on parameters and boundary data.

Application to an optimal control problem.

Abstract

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.