Solutions to degenerative generalized quasilinear Schrödinger equations involving vanishing potentials and critical exponent

In this paper, a class of degenerative quasilinear Schrödinger equations with vanishing potentials and critical Sobolev exponents is considered. The main operator involved in these equations is not strictly elliptic. Under suitable conditions, the existence of nontrivial solutions to the equations is obtained by employing variational methods and the decay rate of the solutions is established.