Existence of positive entire solutions of some semilinear elliptic equations (Q1188220)

Elliptic equations of Emden-Fowler type are considered: \[ \Delta u+ K(x)u^ m,\quad u\in\mathbb{R}^ n,\quad n\geq 3,\quad m\neq 1. \] For \(n=3\), \(K(x)=1/(1+| x|^ 2)\) it represents Matukuma's model of dynamics for globular clusters of stars. The condition \(\int^ \infty_ 0 rK(r)\;dr<\infty\) combined with some conditions at \(r=0\) was shown by Kawano, Satsumo and Yotsutani to be sufficient for the existence of positive solutions. (\(K\) is regarded as a function of the radius \(r\)). The authors show that if \(K\) is continuous and \(K(r)=Ar^ p+o(r^ p)\) at \(r=0\) and \(K(r)=Br^ q+o(r^ q)\) at \(r=\infty\) and some inequalities on \(p\) and \(q\) are true then there exists a positive entire solution. To prove it they consider the behavior of solutions to the equation \(u''+{n- 1\over r} u'+ K(r) u^ m=0\) on separate intervals \(r\in (0,1]\) and \(r\in [1,\infty)\), with conditions at zero for the first problem \((r\in (0,1])\) and at infinity for the second one.