Well-posedness and weak mean attractors of stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian lattice systems with superlinear Lévy noises

This paper is concerned with the well-posedness and weak pullback mean random attractors of stochastic $ p $-Laplacian lattice systems driven by superlinear Lévy noises. We first prove the existence and uniqueness of solutions of the stchastic system. Based on the global well-posedness of the stochastic system, we define a mean random dynamical system via the solution operators, and then prove the existence and uniqueness of the weak pullback mean random attractors of the mean random dynamical system. Several appropriate stopping times are defined to treat the locally Lipschitz continuity of the discrete $ p $-Laplacian operator as well as the nonlinear drift and diffusion terms. Our results are new even when the growth rate of the Lévy noises is reduced to be linear.