Global, small radially symmetric solutions to nonlinear Schrödinger equations and a gauge transformation

Abstract

This paper proves the global existence of small radially symmetric solutions to the nonlinear Schrödinger equations of the form $$ \begin{cases} i\partial_t u+\frac12\triangle u=F(u, \nabla u, \bar u, \nabla\bar u),& (t,x)\in\mathbb{R}\times\mathbb{R}^n,\\ u(0,x)=\varepsilon_0\phi(|x|),& x\in\mathbb{R}^n, \end{cases} $$ where $n\ge 3$, $\varepsilon_0$ is sufficiently small, $|x|=(\sum\limits_{1\le j\le n}x_j^2)^{1/2}$, $$ F =\sum_{\ell_0\le|\alpha|\le\ell_1}\lambda_\alpha u^{\alpha_1}\bar u^{\alpha_2} +\sum_{\substack {0\le|\alpha|\le\ell_2\\ 1\le |\beta|\le\ell_3}} \lambda_{\alpha\beta}u^{\alpha_1}\bar u^{\alpha_2}(\sum_{1\le j\le n}|\partial_ju|^2)^{\beta_1}(\sum_{1\le j\le n}(\partial_ju)^2)^{\beta_2} (\sum_{1\le j\le n}(\partial_j\bar u)^2)^{\beta_3} $$ with $\lambda_\alpha, \lambda_{\alpha\beta}\in\mathbb{C}$, $\ell_1, \ell_2, \ell_3\in\mathbb{N}$, $\ell_0=3$ for $n=3, 4$, and $\ell_0=2$ for $n\ge 5$. The method depends on the combination of a gauge transformation and generalized energy estimtes and does not require the condition such that $$ \partial_{\nabla u}F\ \text{ is pure imaginary} $$ which is needed for the classical energy method.