Positive entire solutions of superlinear elliptic equations (Q1087719)

This paper discusses the existence and nonexistence of radially symmetric, positive entire solutions of \[ \Delta u+p(| x|)u^{\gamma}=0,\quad x\in {\mathbb{R}}^ n \] where \(n\geq 3\) and \(\gamma >1\). It is shown that if \((d/dt)(t^ rp(t))\leq 0\), \(t>0\) then for any \(\alpha >0\) there exists such a solution with \(u(0)=\alpha\). Here \(r=()(n+2-\gamma (n-2))\). Furthermore if \(d/dt(t^ rp(t))\geq 0\), \(t>0\) and \(t^ rp(t)\to \infty\) as \(t\to \infty\), then there are no such solutions.