On a special class of nonlocal Neumann problems with critical growth

In this work, we investigate existence and multiplicity of nontrivial solutions for a nonlocal Neumann problem, with a particular focus on the case where the nonlocal term exhibits a decreasing behavior. Our main results are derived by utilizing the concentration-compactness principle originally introduced by Lions and extended by Bonder, Saintier and Silva. By employing trace theory and the Mountain Pass Theorem, we demonstrate that, under suitable conditions, the problem admits a nontrivial solution, and multiplicity of solutions is obtained via a truncation argument.