Asymptotics of a free-boundary problem (Q1919649)

For the boundary value problem \((*)\) \(u''+{{2\nu+1}\over\nu} u'+u-u^q=0\), \(0<r<R\), \(u'(0)=0\), \(u(R)= u'(R)=0\), the following theorem is proved: For each \(q\in[0,1)\), there is a unique \(R>0\) such that \((*)\) admits a (classical) solution \(u\) that is positive everywhere on \((0,R)\). The function \(u\) is monotonically decreasing on \((0,R)\); \(u(0)\) is bounded, but \(R\) grows beyond as \(q\to1\).