论文标题
双重动力非线性schrödinger方程及其概括：唯一性，非排位和应用
The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications
论文作者
论文摘要
在本文中，我们首先证明了关于$ΔU+g（u）= 0 $的方程式的阳性溶液的独特性和非分类的一般结果。我们的结果特别适用于双功率非线性，其中$ g（u）= u^q-u^p-μu$ for $ p> q> q>>>> 1 $和$μ> 0 $，我们将与更多详细信息进行讨论。在这种情况下，唯一解决方案的非分类$u_μ$使我们能够在两个限制中得出其行为，$μ\ to0 $和$μ\toμ_*$，其中$μ_*$是存在的阈值。这给出了在某些制度中固定质量下能量最小化器的独特性。我们还对$u_μ$的$ l^2 $质量的变化构成$μ$的猜想，我们用数值模拟说明了这一点。如果有效，则此猜想将暗示所有情况下能量最小化器的独特性，还提供有关$U_μ$的轨道稳定性的一些重要信息。
In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $Δu+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-μu$ for $p>q>1$ and $μ>0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_μ$ allows us to derive its behavior in the two limits $μ\to0$ and $μ\toμ_*$ where $μ_*$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the $L^2$ mass of $u_μ$ in terms of $μ$, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of $u_μ$.
