论文标题
具有电势的耦合的非线性毛 - 皮塔维斯基方程的单个峰溶液的唯一性
Uniqueness of Single Peak Solutions for Coupled Nonlinear Gross-Pitaevskii Equations with Potentials
论文作者
论文摘要
对于几个奇异的扰动的GROSS-PITAEVSKII方程，我们首先证明，如果单个峰溶液集中在同一点上，则只要泰勒围绕浓度点的电势膨胀在同一顺序上，就沿着所有方向占据相同的顺序。除其他假设外，我们的结果表明，在[21,31,38]中获得的峰值解决方案是独一无二的。此外，对于径向对称的环形电势，它的最低限度达到$γ_i：= \ {x \ in \ mathbb {r}^n：| x | | | x | = a_i> 0 \}，i = 1,2，\ cdots，l，$，并且在$γ_i$γ_i$γ_i$γ_i的固定范围内完全变成了阳性并且是围绕原点旋转的独特之处。据我们所知，这是径向对称但非单调电位下基态的第一个唯一性结果。
For a couple of singularly perturbed Gross-Pitaevskii equations, we first prove that the single peak solutions, if they concentrate on the same point, are unique provided that the Taylor's expansion of potentials around the concentration point is in the same order along all directions. Among other assumptions, our results indicate that the peak solutions obtained in [21,31,38] are unique. Moreover, for the radially symmetric ring-shaped potential, which attains its minimum at the spheres$Γ_i:=\{x\in\mathbb{R}^N:|x|=A_i>0\},i=1,2,\cdots,l,$ and is totally degenerate in the tangential space of $Γ_i$, we prove that the positive ground state is cylindrically symmetric and is unique up to rotations around the origin. Aa far as we know, this is the first uniqueness result for ground states under radially symmetric but non-monotonic potentials.
