This paper investigates the nonlinear static bending, nonlinear vibration, and thermal postbuckling of functionally graded graphene platelets (FG-GPLs) strengthened sandwich beams with a negative Poisson's ratio core resting on three-parameter nonlinear elastic foundations. The effect of nonlinear hygrothermal conditions on the results is also examined. The face layers are made of copper reinforced with GPLs that are uniformly dispersed or functionally graded through the thickness of the layers. All effective material properties are expressed in terms of the graphene volume fraction, which varies through the thickness according to a refined cosine rule. Accordingly, five patterns of GPL distribution are presented. The auxetic core is made of hexagonal copper cells. A refined shear deformation beam theory is used to derive the governing differential equations within the framework of von Karman's relations. Using the Galerkin method, the equilibrium equations are converted into a nonlinear algebraic system, while the motion equations are converted into nonlinear ordinary differential equations (NODEs). The nonlinear algebraic system is numerically solved using Newton's method, while the NODEs are solved using the fourth-order Runge–Kutta method. Several comparison studies are conducted to examine the accuracy of the present solutions. Moreover, the impacts of different parameters, such as core-to-face thickness ratio, moisture concentration, temperature rise, graphene distribution type, elastic foundation stiffness, and length-to-depth ratio, on the linear and nonlinear responses of FG-GPLs-strengthened sandwich beams with auxetic core are discussed in detail.