The nonlinear Schr\"odinger equation with combined power-type nonlinearities

Abstract

We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and $\lambda_2$ are nonzero real constants, and $0<p_1<p_2\le \frac 4{n-2}$. We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space $H^1(\R^n)$ and in the pseudoconformal space $\Sigma:=\{f\in H^1(\R^n); xf\in L^2(\R^n)\}$. Of particular interest is the case when both nonlinearities are defocusing and correspond to the $L_x^2$-critical, respectively $\dot H^1_x$-critical NLS, that is, $\lambda_1, \lambda_2>0$ and $p_1=\frac{4}{n}$, $p_2=\frac{4}{n-2}$. The results at the endpoint $p_1 = \frac{4}{n}$ are conditional on a conjectured global existence and spacetime estimate for the $L^2_x$-critical nonlinear Schr\"odinger equation. As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in $H^1_x$ for solutions to the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = |u|^{p} u, $$ with $\frac{4}{n}<p<\frac{4}{n-2}$, which was first obtained by J. Ginibre and G. Velo, \cite{gv:scatter}.