Existence and Uniqueness of Solutions to Non-Local Problems of Brézis–Oswald Type and Its Application

The aim of this paper is to establish the existence and uniqueness of solutions to non-local problems involving a discontinuous Kirchhoff-type function via a global minimum principle of Ricceri. More precisely, we first obtain the uniqueness result of weak solutions to nonlinear fractional Laplacian problems of Brézis–Oswald type. We then demonstrate the existence of a unique positive solution to Kirchhoff-type problems driven by the non-local fractional Laplacian as its application. The main features of the present paper are the lack of the continuity of the Kirchhoff function in [0,∞) and the localization of a positive solution.