Existence and Uniqueness of Positive Solutions to Fractional Problems of Brézis–Oswald-Type with Unbalanced Growths and Hardy Potentials

The aim of this paper is to establish the existence and uniqueness of positive solutions to the non-local Brézis–Oswald-type fractional problems that involve fractional (r,q)-Laplace operators and Hardy potentials. In particular, we observe an eigenvalue problem associated with the fractional (r,q)-Laplacian to determine the existence of at least one positive weak solution for our problem. The main features of this paper are the lack of the semicontinuity property of an energy functional related to our problem and the presence of a singular coefficient. The decisive tool for overcoming this technical difficulty is the concentration–compactness principle in fractional, critical and Hardy terms. Moreover, we establish the uniqueness results of Brézis–Oswald–type by exploiting a generalization of the discrete Picone inequality.