This thesis is composed of two independent parts.
In the first part, the Schr\"odinger-Poisson systems which can be used to describe the models arising from the applied sciences are studied. We will focus on two kinds of models. The first one is the bipolar Schr\"odinger-Poisson system for which we consider the wellposedness and large time behavior of solutions to Cauchy problem. The second one is the quasi-linear Schr\"odinger-Poisson system for which we investigate the initial-boundary value problem with the Dirichlet boundary conditions. Firstly, for the Cauchy problem of bipolar Schr\"odinger-Poisson system, we obtain the global wellposedness of mild solution to the Cauchy problem with the initial data in $H^s$ ($s$ is an arbitrary real number) by using the Strichartz estimates and the properties of Besov spaces, which extends and improves the related known results in which $s$ is only a nonnegative integer. Secondly, we obtain the global wellposedness and large time behavior of solutions to the Cauchy problem of the bipolar nonlinear Schr\"odinger-Poisson system with a defocusing self-interacting potential. And we get the energy, pseudo-conformal conservation laws which are valid for all space dimensions. Thirdly, in order to investigate the asymptotic behavior of the solutions for the Cauchy problem to bipolar nonlinear Schr\"odinger-Poisson system, we establish initiatively the theory for the modified scattering operators. The methods used to the bipolar cases can be applied into the unipolar cases without any difficulties. At last, we obtain the existence and uniqueness of the solution to the initial-boundary value problem for the quasi-linear Schr\"odinger-Poisson system with Dirichlet boundary conditions, which holds for one, two and three space dimensions.
In the second part, we study the fourth order nonlinear Schr\"odinger equations with nonlinearities which may contain derivatives in one dimensional space and in multi-dimensional spaces. By utilizing the modern methods in analysis such as local smoothing effect method, the estimates for maximal functions etc., we establish the local wellposedness to this problem. We not only avoid the fetters of classical methods but also overcome the difficulties arising from the higher derivatives. And in the multi-dimensional cases, the focusing and defocusing cases are both considered for the second order derivative term.
Key words: bipolar Schr\"odinger-Poisson system, quasi-linear Schr\"odinger-Poisson system, fourth order nonlinear Schr\"odinger equation, wellposedness, scattering operator