Normalized solutions for critical Schrödinger–Bopp–Podolsky systems with p-Laplacian and competing potentials

The paper deals with the following critical Schrödinger–Bopp–Podolsky system [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text] is the Bopp–Podolsky constant, [Formula: see text] appears as a Lagrange multiplier, [Formula: see text], [Formula: see text] is the Sobolev critical exponent, [Formula: see text] is the [Formula: see text]-Laplace operator, the absorption potential [Formula: see text] and the reaction potential [Formula: see text] are continuous functions, and [Formula: see text] is a continuous function with subcritical growth at infinity. To our best knowledge, this paper is the first study on normalized solutions for a Schrödinger–Bopp–Podolsky system involving the [Formula: see text]-Laplacian, the Sobolev critical exponent and competing potentials. With the aid of minimization techniques and the Lusternik–Schnirelmann category theory, the existence, concentration, multiplicity and nonexistence of normalized solutions for system [Formula: see text] are obtained. To some extent, our main theorems complement and extend the results of Alves and Thin [On existence of multiple normalized solution to a class of elliptic problems in whole [Formula: see text] via Lusternik–Schnirelmann category, SIAM J. Math. Anal. 55 (2023) 1264–1283], D’Avenia and Siciliano [Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: Solutions in the electrostatic case, J. Differential Equations 267 (2019) 1025–1065], Santos et al. [Critical Schrödinger–Bopp–Podolsky systems: solutions in the semiclassical limit, Calc. Var. Partial Differential Equations 63 (2024) 155] and Zhang et al. [Double phase problems with competing potentials: concentration and multiplication of ground states, Math. Z. 301 (2022) 4037–4078].