Existence of positive radial solutions to \(\Delta u+K(| x| )u^ p=0\) in \(\mathbb R^ n\). (English) Zbl 0813.34036

Yanagida, Eiji; Yotsutani, Shoji

J. Differ. Equations 115, No. 2, 477-502 (1995).

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)].

Reviewer: P. Drábek (Plzeň)

MSC:

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

Keywords:

semilinear elliptic problems; positive radial solutions; rapidly decaying solution