Weak Solutions for a Class of Nonlocal Singular Problems Over the Nehari Manifold

In this paper, we consider a nonlocal model of dilatant non-Newtonian fluid with a Dirichlet boundary condition. By using the Nehari manifold and fibering map methods, we obtain the existence of at least two weak solutions, with sign information. In detail, we construct two local minimizers of the energy functional associate to the main problem on proper Nehari submanifolds; in this way, we bypass the singularity to get boundedness below and coercivity of the functional. Then, we conclude that such minimizers correspond to weak solutions.