Year: 2010
Journal of Partial Differential Equations, Vol. 23 (2010), Iss. 4 : pp. 366–373
Abstract
We consider the problem of whether the equation $Δu = p(x) f (u)$ on $R^N, N ≥ 3$, has a positive solution for which $lim_{|x|→∞} u(x)=∞$ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,∞) and satisfies $∫^∞_1[F(t)]^{-1/2}dt=∞$ where $F(t)=∫^t_0f(s)ds$. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies $∫^∞_0r\min_{|x|=r}p(x)dr=∞$. Conversely, we show that a necessary condition for the solution to exist is that p satisfies $∫^∞_0r^{1+ε}\min_{|x|=r}p(x)dr=∞$ for all $ε > 0$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v23.n4.4
Journal of Partial Differential Equations, Vol. 23 (2010), Iss. 4 : pp. 366–373
Published online: 2010-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 8
Keywords: Large solution