In the present work we establish the existence and multiplicity of positive solutions for a critical elliptic problem in the whole space \({\mathbb {R}}^N\) . The main feature here is to treat a Kirchhoff-type elliptic problem where the nonlinearity is critical and defines a sign-changing function. Our approach relies on the minimization method applied to the Nehari manifold together with the nonlinear Rayleigh quotient method. Here, we use the fibering maps associated with the energy functional which exhibits degenerate points under suitable values on the two parameters within the nonlinearity. This difficulty does not allow us to apply the Lagrange Multipliers Theorem in general. Furthermore, our nonlinearity does not satisfy the famous Ambrosetti-Rabinowitz condition. Our main contribution relies on restoring the strong convergence and compactness results from the Sobolev spaces into the Lebesgue spaces. Here, we establish also some nonexistence results under specific assumptions on the nonlinearity by using a Pohozaev identity.
