### Impacto

Herrero, Miguel A. and Velázquez, J.J. L.
(1991)
*Radial solutions of a semilinear elliptic problem.*
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 118
(3-4).
pp. 305-326.
ISSN 0308-2105

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Official URL: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8244935

## Abstract

We analyse the set of nonnegative, global, and radial solutions (radial solutions, for short) of the equation -Δu + u(p) = f in R(N), N ≥ 1, where 0 < p < 1, and f element-of L(loc)1(R(N)) is a radial and almost everywhere nonnegative function. We show that radial solutions of (E) exist if f(r) = o(r2p/1-p) or if f(r) ≈ cr2p/1-p as r --> ∞, where [GRAPHICS] When f(r) = c*r2p/1-p + h(r) with h(r) = o(r2p/1-p) as r --> ∞, radial solutions continue to exist if h(r) is sufficiently small at infinity. Existence, however, breaks down if h(r) > 0, [GRAPHICS] Whenever they exist, radial solutions are characterised in terms of their asymptotic behaviour as r --> ∞.

Item Type: | Article |
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Uncontrolled Keywords: | Equation; RN; set of nonnegative; global and radial solutions |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 17132 |

References: | R. Bellman. Stability theory of differential equations (New York: Dover, 1953). H. Brezis. Semilinear equations in RN without conditions at infinity. Appl. Math. Optim. 12 (1984), 271-282. T. Gallouët and J. M. Morel. The equation -Δu + |u|α-1u = f for 0 ≤ α ≤ 1. J. Nonlinear Anal. 11 (1987), 893-912. M. A. Herrero and J. J. L. Velázquez. On the dynamics of a semilinear heat equation with strong absorption. Comm. Partial Differential Equations 14 (1989), 1653-1715. M. Murata. Structure of positive solutions to (-Δ + V)u=0 in RN. Duke Math J. 53 (1986), 869-943. |

Deposited On: | 20 Nov 2012 12:41 |

Last Modified: | 07 Feb 2014 09:42 |

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