Existence and nonexistence of radial solutions for semilinear equations with bounded nonlinearities on exterior domains

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Article studies radial solutions of ∆u + K(r)f(u) = 0 on the exterior of the ball of radius R > 0 centered at the origin in ℝᶰ where f is odd with f < 0 on (0, β), f > 0 on (β, δ), f ≡ 0 for u > δ, and where the function K(r) is assumed to be positive and K(r) → 0 as r → ∞.

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16 p.

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Iaia, Joseph A. December 1, 2020.

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Article studies radial solutions of ∆u + K(r)f(u) = 0 on the exterior of the ball of radius R > 0 centered at the origin in ℝᶰ where f is odd with f < 0 on (0, β), f > 0 on (β, δ), f ≡ 0 for u > δ, and where the function K(r) is assumed to be positive and K(r) → 0 as r → ∞.

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16 p.

Notes

Abstract: In this article we study radial solutions of ∆u + K(r)f(u) = 0 on the exterior of the ball of radius R > 0 centered at the origin in ℝᶰ where f is odd with f < 0 on (0, β), f > 0 on (β, δ), f ≡ 0 for u > δ, and where the function K(r) is assumed to be positive and K(r) → 0 as r → ∞. The primitive F(u) = ∫₀ᵘ f(t) dt has a “hilltop” at u = δ. With mild assumptions on f we prove that if K(r) ∼ r ⁻ᵅ with 2 < α < 2(N − 1) then there are n solutions of ∆u + K(r)f(u) = 0 on the exterior of the ball of radius R such that u → 0 as r → ∞ if R > 0 is sufficiently small. We also show there are no solutions if R > 0 is sufficiently large.

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  • Electronic Journal of Differential Equations, 2020(117), Texas State University. Department of Mathematics., December 1, 2020, pp. 1-16

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  • Publication Title: Electronic Journal of Differential Equations
  • Volume: 2020
  • Article Identifier: 117
  • Peer Reviewed: Yes

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  • December 1, 2020

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  • May 27, 2022, 5:55 a.m.

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  • Nov. 28, 2023, 2:19 p.m.

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Iaia, Joseph A. Existence and nonexistence of radial solutions for semilinear equations with bounded nonlinearities on exterior domains, article, December 1, 2020; (https://digital.library.unt.edu/ark:/67531/metadc1934130/: accessed April 14, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Science.

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