Isolated singularities of the capillary equation with negative gravity

We classify isolated singularities of classical solutions to the capillary equation with negative gravity, \[ \operatorname{div}\frac{Du}{\sqrt{1+|Du|^2}}=-u \qquad\text{in }B_R\setminus\{0\}\subset\mathbb R^n, \qquad n\ge2, \] without a priori assumptions on symmetry, sign, one-sided boundedness, or blow-up rate. Every such solution is either bounded near the puncture or tends uniformly to $+\infty$ or $-\infty$. In the bounded case, the solution extends across the puncture as a $W^{1,1}_{\mathrm{loc}}$ distributional solution; this extension has a unique continuous representative when $2\le n\le7$. In the unbounded case, for some $\varepsilon\in\{-1,1\}$, \[ u(x)=\varepsilon\left(\frac{n-1}{|x|} -\frac{n+3}{2(n-1)^2}|x|^3\right)+O(|x|^5), \] uniformly in the angular variable. In every fixed smaller ball, all sufficiently high level sets of $\varepsilon u$ are connected, smooth, strictly convex hypersurfaces enclosing the puncture. The corresponding pressure-rescaled graphs converge smoothly with multiplicity one to the round cylinder $\partial B_{n-1}\times\mathbb R$. Nonradial examples form infinite-dimensional families that agree with the radial pole to every algebraic order, so the complete asymptotic expansion does not determine the singular solution germ. The proof of the classification combines critical tail estimates and logarithmic $BV$ compactness with level-set rigidity and the translation identities of the capillary equation. The critical two-dimensional case requires an additional finite-height analysis to exclude a translation defect.