We establish the existence of a least-energy sign-changing solution for a Schrödinger-Bopp-Podolsky system in \(\mathbb {R}^3\) . The problem is characterized by the presence of a critical Sobolev nonlinearity and a non-coercive steep potential well. The proof relies on variational methods, specifically analyzing the associated energy functional restricted to the sign-changing Nehari manifold. A primary difficulty-the lack of compactness arising from the critical exponent-is overcome through a delicate energy analysis combined with Lions’ concentration-compactness principle. We demonstrate that for a sufficiently deep potential, a nodal ground state solution exists which changes sign exactly once. Furthermore, its energy is shown to be strictly greater than twice the energy of the least-energy solutions. Our result is the first to address this problem under the combined setting of critical growth and a steep potential, extending previous works that assumed either coercive potentials or more restrictive subcritical nonlinearities.
