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Existence of positive radial solutions to $$\Delta u+K(| x| )u^ p=0$$ in $$\mathbb R^ n$$. (English) Zbl 0813.34036
The existence of positive radial solutions to the equation $$\Delta u + K (| x |) u^ P = 0$$, $$x \in \mathbb R^ n$$, where $$p>1$$, $$n>2$$, is considered. The following conditions are imposed on $$K(r)$$: $$K(r)$$ is continuous on $$(0, \infty)$$; $$K(r) \geq 0$$ and $$K(r) \not \equiv 0$$ on $$(0,\infty)$$; $$rK(r) \in L^ 1(0,1)$$; $$r^{n - 1 - (n - 2)p} K(r) \in L^ 1(1,\infty)$$. Radial solutions are classified as crossing solution, slowly decaying solution and rapidly decaying solution. The main theorem concerns the existence of rapidly decaying solutions and completes results of previous work [N. Kawano and the authors, Funkc. Ekvacioj 36, 557–579 (1993; Zbl 0793.34024)].

##### MSC:
 34C11 Growth and boundedness of solutions to ordinary differential equations 35J60 Nonlinear elliptic equations 35J15 Second-order elliptic equations
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