Rotationally symmetric hypersurfaces with prescribed mean curvature
The physical problem of characterizing the shape of a rotationally symmetric pendent liquid drop in a gravity field leads to that of finding symmetric solutions to the equation \[ \text{div } {{Du} \over {\sqrt {1+|Du|^2}}}+ \kappa u=0, \qquad \kappa>0, \tag{1} \] for the height \(u(r)\) of the surface near the tip \(u_0= u(0)\) of the drop. \textit{P. Concus} and the reviewer [Invent. Math. 29, 143-148 (1975; Zbl 0319.76007)] showed the local existence of a solution of (1) with an isolated singularity, and obtained an asymptotic expansion valid for any such solution. Later they showed that for large negative \(u_0\) there exists a globally regular solution \(u(r)\) of an associated parametric system, yielding many nearly spherical ``bubbles'' along the negative \(u\)-axis, then becoming a graph \(u(r)\) that oscillates about the \(r\)-axis to infinity. The reviewer proved that a subsequence of these solutions converges, uniformly in compacta, to a solution of (1) with an isolated singularity and the same (divergent) asymptotic expansion. In the present work, the author extends and in some ways improves those results, corresponding to equations of the general form \[ \text{div } {{Du} \over {\sqrt {1+|Du|^2}}}+ f(u)= 0 \tag{2} \] in \(\mathbb{R}^N\), under suitable hypotheses on \(f(u)\). The extensions are of considerable interest and decidedly nontrivial. She obtains a number of explicit estimates, yielding conditions for existence of singular solutions and characterizing the asymptotic and global behavior of such solutions when they exist. Her results are obtained in part via procedures analogous to those of the earlier papers, but in extending them to the present context she makes also effective use of subtle energy relations, connected with identities of Pohozaev type, that did not enter into the earlier contributions. It seems worth remarking that if \(f' (u)\leq 0\), then every isolated singularity of a solution of (2) is removable. Thus, it is clear that the global behavior of solutions is very strongly dependent on the choice of \(f(u)\), and it has to be expected that the particular growth hypotheses introduced by the author cannot be greatly weakened for the results obtained. The question of uniqueness of the singular solutions, conjectured by the reviewer in the particular case (1), remains open, as the author points out.
scientific article; zbMATH DE number 910275