Existence and Nonexistence of Solutions for Sublinear Equations on Exterior Domains

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This article studies radial solutions of Δu + K(r)ƒ(u) = 0 on the exterior of the ball of radius R > 0, BR, centered at the origin in ℝN with u = 0 on @BR where ƒ is odd with ƒ < 0 on (0; β), ƒ > 0 on (β;∞), f(u) ~ uᵖ with 0 < p < 1 for large u and K(r) ~ r⁻ᵅ for large r.

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Iaia, Joseph A. September 13, 2017.

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This article studies radial solutions of Δu + K(r)ƒ(u) = 0 on the exterior of the ball of radius R > 0, BR, centered at the origin in ℝN with u = 0 on @BR where ƒ is odd with ƒ < 0 on (0; β), ƒ > 0 on (β;∞),
f(u) ~ uᵖ with 0 < p < 1 for large u and K(r) ~ r⁻ᵅ for large r.

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12 p.

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Abstract: In this article we study radial solutions of Δu + K(r)ƒ(u) = 0 on the exterior of the ball of radius R > 0, BR, centered at the origin in ℝN with u = 0 on @BR where ƒ is odd with ƒ < 0 on (0; β), ƒ > 0 on (β;∞),
f(u) ~ uᵖ with 0 < p < 1 for large u and K(r) ~ r⁻ᵅ for large r. We prove that if N > 2 and K(r) ~ r⁻ᵅ with 2 < α < 2(N - 1) then there are no solutions with limᵣ⃯∞u(r) = 0 for sufficiently large R > 0. On the other hand, if 2 < N - p(N - 2) < α < 2(N - 1) and k, n are nonnegative integers with 0 ≤ k ≤ n then there exist solutions, uK, with k zeros on (R,∞) and limᵣ⃯∞u(r) = 0 if R > 0 is sufficiently small.

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  • Electronic Journal of Differential Equations, 2017(214), Texas State University, September 13, 2017, pp. 1-12

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  • Publication Title: Electronic Journal of Differential Equations
  • Volume: 2017
  • Issue: 214
  • Peer Reviewed: Yes

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  • September 13, 2017

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  • June 15, 2018, 10:41 p.m.

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  • Nov. 28, 2023, 2:14 p.m.

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Iaia, Joseph A. Existence and Nonexistence of Solutions for Sublinear Equations on Exterior Domains, article, September 13, 2017; San Marcos, Texas. (https://digital.library.unt.edu/ark:/67531/metadc1164557/: accessed December 4, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Science.

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