LOCAL EXISTENCE OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEAR SCHRÖDINGER EQUATIONS

Abstract

In this paper we consider the local existence to the Cauchy problem for nonlinear Schrödinger equations with power nonlinearities

$$(*)\{\begin{array}{l}i{\partial }_{t}u+\frac{1}{2}\mathrm{\Delta }u=\mathcal{N}(u,\nabla u,\overline{u},\nabla \overline{u}),(t,x)\in \mathbf{R}\times {\mathbf{R}}^n,\hfill \\ u(0,x)=u_0(x),x\in {\mathbf{R}}^n,\hfill \end{array}$$

where $n\ge 2$ and

$$\mathcal{N}=\mathcal{N}(u,w,\overline{u},\overline{w})={\displaystyle \sum }_{l_0\le \left|\alpha \right|+\left|\beta \right|+\left|\gamma \right|\le l_1}{\lambda }_{\alpha \beta \gamma }{u}^{{\alpha }_1}{\overline{u}}^{{\alpha }_2}{\displaystyle \prod }_{j=1}^n{(w_j)}^{{\beta }_j}{\displaystyle \prod }_{k=1}^n{({\overline{w}}_k)}^{{\gamma }_k}$$

With $w={\left(w_j\right)}_{1\le j\le n},{\text{λ}}_{\alpha \beta \gamma }\in \mathbf{\text{C}},l_0\in \mathbf{\text{N}},l_1,l_0\ge 2$. Classical energy method is useful to show local existence in time of solutions to $(*)$ when ${\partial }_w\mathcal{N}$ is pure imaginary (see, [10, 14-16]), and in this case it is known that there exists a unique solution if $u_0\in H^{\left[\frac{n}{2}\right]+3,0}$ (see [10]), where . However, if ${\partial }_w\mathcal{N}$ is not pure imaginary, there are only a few results [2,12,13] that require higher order Sobolev spaces compared with [10, 14-16] because the classical energy method does not work for the problem. Our purpose in this paper is to show local existence in time of solutions to $\text{RR} :\frac{A,Г\to B}{\square A,\square Г\to \square B}$ in the weighted Sobolev space $H^{\left[\frac{n}{2}\right]+6,0}{\cap }^{\text{}}{H}^{\left[\frac{n}{2}\right]+3,2}$ without any size restriction on the data. Our function spaces are more natural than those used in [2,12,13].