三维空间中$-Δu+u-u^3=0$激发态的唯一性
Uniqueness of excited states to $-Δu+u-u^3=0$ in three dimensions
AI总结：
针对三维非线性椭圆方程$-Δu+u-u^3=0$的径向激发态唯一性问题，将对应常微分方程视为双势阱阻尼振子，通过严格数值分析解的能量与变分，验证相关不等式从而证明了若干激发态的唯一性。
摘要：
我们证明了针对模型非线性项$f(y) = y^3 - y$，常微分方程$\begin{cases}\begin{aligned}\frac{d^2 y}{dt^2}(t) + \frac{2}{t}\frac{dy}{dt}(t) + f(y(t)) &= 0 \\ y(0) &= b \\ \frac{dy}{dt}(0) &= 0 \end{aligned}\end{cases}$的若干激发态具有唯一性。第$n$激发态指恰好有$n$个零点、且当$t \to \infty$时趋于0的解，它们对应偏微分方程$Δu + f(u)= 0$在$H^1$空间中的所有光滑径向非零解。我们将该常微分方程视为受双势阱作用的阻尼振子，通过对解的能量与变分进行严格数值分析证明了上述结果。具体而言，唯一性问题可完全转化为关于解及其变分的不等式问题，且这些不等式可通过数值方法验证。
英文摘要：
We prove the uniqueness of several excited states to the ODE $\ddot y(t) + \frac{2}{t} \dot y(t) + f(y(t)) = 0$, $y(0) = b$, and $\dot y(0) = 0$ for the model nonlinearity $f(y) = y^3 - y$. The $n$-th excited state is a solution with exactly $n$ zeros and which tends to $0$ as $t \to \infty$. These represent all smooth radial nonzero solutions to the PDE $Δu + f(u)= 0$ in $H^1$. We interpret the ODE as a damped oscillator governed by a double-well potential, and the result is proved via rigorous numerical analysis of the energy and variation of the solutions. More specifically, the problem of uniqueness can be formulated entirely in terms of inequalities on the solutions and their variation, and these inequalities can be verified numerically.
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