Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices

Abstract

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 $\beta_k^{\log}(R)$ is a strictly decreasing $C^1$ bijection from $(0,\infty)$ onto $(\alpha_k^{\log},\infty)$ and satisfies \[ \beta_k^{\log}(R)\longrightarrow\alpha_k^{\log} \quad(R\to\infty),\qquad \log\bigl(\beta_k^{\log}(R)^2\bigr) =\frac{\rho_{k+1}^2}{R^2}+\kappa_{k+1,n}+o(1) \quad(R\downarrow0), \] where $\kappa_{k+1,n}>0$ is given by an explicit Bessel integral. For the sublinear problem, the maximal simple-zero branch exists precisely for $R\in(\rho_{k+1},S_k)$, where $\rho_{k+1}$ is the $(k+1)$-st positive zero of the regular Bessel profile $\Phi_n$, and $S_k$ is the support radius of the unique compactly supported $k$-node whole-space bound state.