Abstract
We study the structure of entire radial solutions of a biharmonic equation with exponential nonlinearity:
with λ = 8(N − 2)(N − 4). It is known from a recent interesting paper by Arioli et al. that (0.1) admits a singular solution U s (r) = ln r −4. We show that for 5 ≤ N ≤ 12, any regular entire radial solution u with u(r) − ln r −4 → 0 as r → ∞ of (0.1) intersects with U s (r) infinitely many times. On the other hand, if N ≥ 13, then u(r) < U s (r) for all r > 0, and the solutions are strictly ordered with respect to the initial value a = u(0). Moreover, the asymptotic expansions of the entire radial solutions near ∞ are also obtained. Our main results give a positive answer to a conjecture in Arioli et al. (J Differ Equ 230:743–770, 2006) [see lines −11 to −9, p. 747 of Arioli et al. (J Differ Equ 230:743–770, 2006)].
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The author thanks the referee for his/her valuable suggestions that improve the presentation of this paper. Research of the author is supported by NSFC (11171092, 10871060) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (114200510011).
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Guo, Z. Further study of entire radial solutions of a biharmonic equation with exponential nonlinearity. Annali di Matematica 193, 187–201 (2014). https://doi.org/10.1007/s10231-012-0272-z
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DOI: https://doi.org/10.1007/s10231-012-0272-z