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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.

MSC:

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
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