Sobolev regularity solutions for a class of singular quasilinear ODEs (Q2063262)

In this paper, the authors consider an initial-boundary value problem for a class of singular quasilinear second order ordinary differential equations with the constraint condition stemming from fluid mechanics, which takes the form \[ u_{r}+\frac{2}{r}u+C^{-1}\left[(2\mu+\lambda)r^{2}u_{rr}+(2r(2\mu+\lambda)-C)u_{r} -2(2\mu+\lambda)u\right]u^{2}+uf(r)=0,\quad r\in(0,1], \] with the initial-boundary condition \[ 0<u(0)=u_{0}<\varepsilon, \quad u(1)=0, \] and the constraint condition \[ \int_0^1 u^{-1}(r)\mathrm{d}r=\overline{C}M, \quad \text{with a fixed constant } \overline{C}, \] where \(0<\varepsilon<1\) and \(\mu, \lambda, C>0\), \(f(r)\) denotes an external force. By means of a suitable Nash-Moser iteration scheme, the authors prove the existence of positive Sobolev regular solutions for this kind of singular quasilinear ODEs. Meanwhile, asymptotic expansion of those positive solutions is shown.