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Asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth

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Díaz Díaz, Jesús Ildefonso and Lazzo, M. and Schmidt, P.G. (2014) Asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth. Journal of Differential Equations, 257 (12). pp. 4249-4276. ISSN 0022-0396

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022039614003350


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Abstract

This paper concerns the blow-up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights. Specifically, we consider radially symmetric solutions of mu = c(|x|)|u| p on an annulus {x ∈ Rn | σ ≤ |x| < ρ}, with ρ ∈ (0,∞) and σ ∈ [0, ρ), that diverge to infinity as |x| → ρ. Here n,m ∈ N, p ∈ (1,∞), and c is a positive continuous function on the interval [σ, ρ]. Letting φρ(r) := QCρ/(ρ −r)q for r ∈ [σ, ρ), with q := 2m/(p−1), Q := (q(q +1)···(q +2m−1))1/(p−1), and Cρ := c(ρ)−1/(p−1), we show that, as |x| → ρ, the ratio u(x)/φρ(|x|) remains between positive constants that depend only on m and p. Extending well-known results for the second-order problem, we prove in the fourth-order case that u(x)/φρ(|x|) → 1 as |x| → ρ and obtain precise asymptotic expansions if c is sufficiently smooth at ρ. In certain higher-order cases, we find solutions for which the ratio u(x)/φρ(|x|)does not converge, but oscillates about 1 with non-vanishing amplitude.


Item Type:Article
Uncontrolled Keywords:Higher-order elliptic equations; Polyharmonic equations; Radial solutions; Large solutions; Boundary blow-up; Asymptotic behavior
Subjects:Sciences > Mathematics > Differential equations
ID Code:28772
Deposited On:26 Feb 2015 07:53
Last Modified:12 Dec 2018 15:06

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