Uniqueness of cylindrically symmetric solutions for coupled Gross-Pitaevskii equations with totally degenerate potentials

Abstract

For a couple of singularly perturbed Gross-Pitaevskii equations, we prove that the single peak solutions concentrating at the same point are unique provided that the Taylor's expansion of potentials $V_1\left(x\right),V_2\left(x\right)$ around the concentration point has the same order along all directions. Under suitable conditions, our results imply that the peak solutions obtained in [21], [31], [38] are unique. Moreover, if the radially symmetric ring-shaped potential attains its minimum at some spheres ${\Gamma }_j:=\{x\in R^N:|x|=A_j>0\},j=1,2,\cdots ,l$, and is totally degenerate in the tangential space of ${\Gamma }_i$, we prove that the positive ground state is cylindrically symmetric and unique up to rotations around the origin. As far as we know, this seems to be the first uniqueness result for ground states under radially symmetric but non-monotonic potentials.