Existence and symmetry of positive ground states for a doubly critical Schrodinger system
arXiv:1211.1514
Abstract
We study the following doubly critical Schrödinger system $$-Δu -\frac{\la_1}{|x|^2}u=u^{2^\ast-1}+ ν\al u^{\al-1}v^\bb, \quad x\in \RN, -Δv -\frac{\la_2}{|x|^2}v=v^{2^\ast-1} + ν\bb u^{\al}v^{\bb-1}, \quad x\in \RN, u, v\in D^{1, 2}(\RN),\quad u, v>0 in $\RN\setminus{0}$},$$ where , $\la_1, \la_2\in (0, \frac{(N-2)^2}{4})$, and $\al>1, \bb>1$ satisfying $\al+\bb=2^\ast$. This problem is related to coupled nonlinear Schrödinger equations with critical exponent for Bose-Einstein condensate. For different ranges of , $\al$, $\bb$ and , we obtain positive ground state solutions via some quite different methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among $\al, \bb$ and 2. Besides, for sufficiently small , positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition can not hold for any positive energy level, which makes the study via variational methods rather complicated.
Final version, to appear in Trans. Amer. Math. Soc