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