This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schrodinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation i∂tu + ǫ�u+δAu+λ|u| � u = 0, x 2 R n , t 2 R, where A represents either the operator � 2 (isotropic dispersion) or Pd=1 ∂xixixixi , 1 � d < n (anisotropic dispersion), and α,ǫ,λ are given real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, such as weak-L p spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation (ǫ = 0) for which, the existence of self-similar solutions is obtained as consequence of his scaling invariance. In a second part, we investigate the vanishing second order dispersion limit in the framework of weak-L p spaces. We also analyze the convergence of the solutions for the nonlinear fourth-order Schrodinger equation i∂tu + ǫ�u + δ� 2 u + λ|u| � u = 0, as ǫ goes to zero, in H 2 -norm, to the solutions of the corresponding biharmonic equation i∂tu + δ� 2 u + λ|u| � u = 0.