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 $k\ge 0$, 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 \[ β_k^{\log}(R)\longrightarrowα_k^{\log} \quad(R\to\infty),\qquad \log\bigl(β_k^{\log}(R)^2\bigr) =\frac{ρ_{k+1}^2}{R^2}+κ_{k+1,n}+o(1) \quad(R\downarrow0), \] where ${\kappa }_{k+1,n}>0$ 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.