Critical degenerate elliptic problems with Grushin operator: Existence, concentration and asymptotic properties of normalized solutions

This paper investigates the following critical degenerate elliptic equation involving the Grushin operator with a prescribed [Formula: see text]-mass constraint: [Formula: see text] where [Formula: see text] is called the Grushin operator, the parameters [Formula: see text] and [Formula: see text], [Formula: see text] with [Formula: see text]. The unknown Lagrange multiplier is denoted by [Formula: see text]. The critical Sobolev exponent is given by [Formula: see text] and the mass-critical exponent is [Formula: see text], and [Formula: see text] represents the homogeneous dimension related to the Grushin operator. Under suitable assumptions on the potential [Formula: see text], we establish the existence of multiple normalized solutions that concentrate at the global minima of [Formula: see text] as [Formula: see text]. We also describe the asymptotic behavior of solutions as [Formula: see text] and show that the limit functions solve the corresponding limit equation and retain concentration properties. In addition, we fully characterize the small-mass asymptotics as [Formula: see text], prove the existence of infinitely many normalized solutions. The proofs rely on variational methods, the concentration-compactness principle and genus theory. Our results significantly extend the theory of normalized solutions for Grushin-type elliptic equations and clarify how potentials and parameters affect the concentration of solutions.