Uniqueness of ground states for the L^2-critical boson star equation

Details Uniqueness of ground states for the L^2-critical boson star equation Uniqueness of ground states for the L^2-critical boson star equation

2009-05-19

arxiv.org/abs/0905.3105v1

Mathematics

Analysis of PDEs

Research announcement note; submitted for publication

Scientific paper

We establish uniqueness of ground states $u(x) \geq 0$ for the $L^2$-critical boson star equation $\sqrt{-\Delta} u - (|x|^{-1} \ast |u|^2) u = -u$ in $\R^3$. The proof blends variational arguments with the harmonic extension to the halfspace $\R^4_+$. Apart from uniqueness, we also show radiality of ground states (up to translations) and the nondegeneracy of the linearization. Our results provide an indispensable basis for the blowup analysis of the time-dependent $L^2$-critical boson star equation. The uniqueness proof can be generalized to different fractional Laplacians $(-\Delta)^s$ and space dimensions.

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