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Radial solutions of a semilinear elliptic problem

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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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
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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