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 (−∆) α/2 for α ∈ (1, 2) and some superlinear and subcritical nonlinearity Gz 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 Painlevé-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.

Lagrange problem for fractional ordinary elliptic system via Dubovitskii–Milyutin method

“…The variational "min" structure of (2) is used in [2]. Similar results are presented in papers [4] and [3] for the control systems possessing the "minimax" and "mountain pass" structures, respectively. For some results concerning the existence, uniqueness, stability and sensitivity of a solution to problem (1), we refer to [2][3][4]14].…”

“…Published by Vilnius University Press This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. problem), hydrodynamics (fractional Navier-Stokes equation and model of the flow in porous media); see [2][3][4] and references therein. System (1) can be viewed as a generalization of the Poisson equation.…”

Symmetry of solutions to nonlocal nonlinear boundary value problems in radial sets

“…There have been several results on the analysis of minimizers involving nonlocal equations, we refer e.g. to [2,[32][33][34]. However, up to the authors knowledge there have been no symmetry results such as the one above.…”

“…As an application we will analyze global minimizers of functionals of the form [2,[32][33][34]. However, up to the authors knowledge there have been no symmetry results such as the one above.…”