奇异椭圆方程的径向节点Dirichlet解：全局分支、端点渐近与Morse指标
Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices
- School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
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AI总结：
本文通过有限球射击分类，证明了对数及次线性奇异椭圆方程径向节点Dirichlet解的唯一性、全局分支、端点渐近与Morse指标，并给出显式Bessel积分常数。
AI中文摘要：
我们为对数方程以及次线性标量场方程在最大简单零点射击类中建立了径向节点Dirichlet解的尖锐有限球射击分类。最近的全空间唯一性和相变定理被用作外部输入，而有限球零曲线范围、端点渐近和谱结论在此证明。这两个模型需要不同的奇异分析：在对数问题中，非线性在节点零点处不是局部Lipschitz的，且线性化势在该处发散；而在次线性问题中，极限全空间轮廓在有限支撑半径处达到二重零点，超出该半径延续不唯一。对于每个$R>0$和$k\ge0$，对数问题具有（至多相差符号）唯一的径向Dirichlet解，且恰好有$k$个内部零点。其射击高度$\beta_k^{\log}(R)$是从$(0,\infty)$到$(\alpha_k^{\log},\infty)$的严格递减$C^1$双射，并满足\\[ \beta_k^{\log}(R)\longrightarrow\alpha_k^{\log} \quad(R\to\infty),\qquad \log\bigl(\beta_k^{\log}(R)^2\bigr) =\frac{\rho_{k+1}^2}{R^2}+\kappa_{k+1,n}+o(1) \quad(R\downarrow0), \\]其中$\kappa_{k+1,n}>0$由显式Bessel积分给出。对于次线性问题，最大简单零点分支恰好在$R\in(\rho_{k+1},S_k)$时存在，其中$\rho_{k+1}$是正则Bessel轮廓$\Phi_n$的第$(k+1)$个正零点，$S_k$是唯一紧支撑$k$节点全空间束缚态的支撑半径。
英文摘要：
We establish sharp finite-ball shooting classifications for radial nodal Dirichlet solutions of the logarithmic equation and, within the maximal simple-zero shooting class, of the sublinear scalar-field equation. Recent whole-space uniqueness and phase-transition theorems are used as external inputs, while the finite-ball zero-curve ranges, endpoint asymptotics, and spectral consequences are proved here. The two models require different singular analyses: in the logarithmic problem the nonlinearity is not locally Lipschitz at a nodal zero and the linearized potential diverges there, whereas in the sublinear problem the limiting whole-space profile reaches a double zero at a finite support radius, beyond which continuation is nonunique. For every $R>0$ and $k\ge0$, the logarithmic problem has, up to sign, a unique radial Dirichlet solution with exactly $k$ interior zeros. Its shooting height $β_k^{\log}(R)$ is a strictly decreasing $C^1$ bijection from $(0,\infty)$ onto $(α_k^{\log},\infty)$ and satisfies \[ β_k^{\log}(R)\longrightarrowα_k^{\log} \quad(R\to\infty),\qquad \log\bigl(β_k^{\log}(R)^2\bigr) =\frac{ρ_{k+1}^2}{R^2}+κ_{k+1,n}+o(1) \quad(R\downarrow0), \] where $κ_{k+1,n}>0$ is given by an explicit Bessel integral. For the sublinear problem, the maximal simple-zero branch exists precisely for $R\in(ρ_{k+1},S_k)$, where $ρ_{k+1}$ is the $(k+1)$-st positive zero of the regular Bessel profile $Φ_n$, and $S_k$ is the support radius of the unique compactly supported $k$-node whole-space bound state.
