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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)].

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