Unbounded positive entire solutions of rotationally symmetric harmonic map equations (Q1975326)

The author continues his presentation of results on the rotationally symmetric harmonic map equation \[ \alpha''(r)+(n-1) {f'(r)\over f(r)} \alpha'(r)-(n-1) {g\bigl(\alpha (r)\bigr)g' \bigl(\alpha (r)\bigr)\over f(r)}=0 \] for \(r\in \mathbb{R}^+\), with prescribed limit \(\lim_{r\to 0^+}\alpha (r) =0\), with \(n\in\mathbb{N}\). A basic aim of the author is under certain suitable conditions on \(f\) and \(g\), to construct an unbounded positive \(C^2\)-solution to the problem. By using this existence result, it is shown here that the associated Dirichlet problem at infinity has a unique \(C^2\)-solution, for any nonnegative boundary value at infinity.