Structure theorems for positive radial solutions to \(\Delta u+ K(| x|)u^p=0\) in \(\mathbb R^ n\). (English) Zbl 0793.34024

Let us consider semilinear elliptic equation (1) \(\Delta u+K (| x |) u^ p=0\), \(x \in \mathbb R^n\), where \(p>1\), \(n>2\), \(\Delta=\sum^ n_{i=1} {\partial^ 2 \over \partial x^ 2_ i}\), \(| x |=(\sum^ n_{i=1} x^ 2_ i)^{1/2}\), \(rK(r) \in L^ 1(0,1)\). The structure of positive radial solutions to (1) is studied and it is proved that every such solution is one of the following types: crossing (changing sign) solution, slowly decaying positive solution, rapidly decaying positive solution.


34C11 Growth and boundedness of solutions to ordinary differential equations
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs