Existence of Positive Solutions for the Schrödinger–Bopp–Podolsky Equations

ABSTRACT This paper is concerned with the existence of positive solutions for a class of Schrödinger‐Bopp‐Podolsky systems in three‐dimensional Euclidean space. The system couples a nonlinear Schrödinger equation with a fourth‐order elliptic equation governing the electrostatic potential. We establish existence under assumptions on an external potential, which is allowed to vanish at infinity, and on a nonlinearity with subcritical growth. A key feature of our work is demonstrating that existence hinges on a delicate interplay between the decay rate of the potential and the local behavior of the nonlinearity near the origin. The proof relies on variational methods. We employ a penalization technique to overcome the lack of compactness, establish the existence of a solution for the resulting auxiliary problem via the Mountain Pass Theorem, and use crucial ‐estimates to show this solution solves the original system for a suitable range of parameters.