# A note on boundary blow-up problem of ∆u = u p

Research output: Contribution to journalArticle

### Abstract

Assume that Ω is a bounded domain in R n with n ≥ 2. We study positive solutions to the problem, ∆u = u p in Ω, u(x) → ∞ as x → ∂Ω, where p > 1. Such solutions are called boundary blow-up solutions of ∆u = u p . We show that a boundary blow-up solution exists in any bounded domain if 1 < p (Formula presented). In particular, when n = 2, there exists a boundary blow-up solution to ∆u = u p for all p ∈ (1, ∞). We also prove the uniqueness under the additional assumption that the domain satisfies the condition ∂Ω = ∂Ω.

Original language English 245-251 7 Bulletin of the Korean Mathematical Society 56 1 https://doi.org/10.4134/BKMS.b180221 Published - 2019 Jan 1

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Boundary Blow-up
Blow-up Solution
Bounded Domain
Positive Solution
Uniqueness

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cite this

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title = "A note on boundary blow-up problem of ∆u = u p",
abstract = "Assume that Ω is a bounded domain in R n with n ≥ 2. We study positive solutions to the problem, ∆u = u p in Ω, u(x) → ∞ as x → ∂Ω, where p > 1. Such solutions are called boundary blow-up solutions of ∆u = u p . We show that a boundary blow-up solution exists in any bounded domain if 1 < p (Formula presented). In particular, when n = 2, there exists a boundary blow-up solution to ∆u = u p for all p ∈ (1, ∞). We also prove the uniqueness under the additional assumption that the domain satisfies the condition ∂Ω = ∂Ω.",
author = "Seick Kim",
year = "2019",
month = "1",
day = "1",
doi = "10.4134/BKMS.b180221",
language = "English",
volume = "56",
pages = "245--251",
journal = "Bulletin of the Korean Mathematical Society",
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publisher = "Korean Mathematical Society",
number = "1",

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In: Bulletin of the Korean Mathematical Society, Vol. 56, No. 1, 01.01.2019, p. 245-251.

Research output: Contribution to journalArticle

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N2 - Assume that Ω is a bounded domain in R n with n ≥ 2. We study positive solutions to the problem, ∆u = u p in Ω, u(x) → ∞ as x → ∂Ω, where p > 1. Such solutions are called boundary blow-up solutions of ∆u = u p . We show that a boundary blow-up solution exists in any bounded domain if 1 < p (Formula presented). In particular, when n = 2, there exists a boundary blow-up solution to ∆u = u p for all p ∈ (1, ∞). We also prove the uniqueness under the additional assumption that the domain satisfies the condition ∂Ω = ∂Ω.

AB - Assume that Ω is a bounded domain in R n with n ≥ 2. We study positive solutions to the problem, ∆u = u p in Ω, u(x) → ∞ as x → ∂Ω, where p > 1. Such solutions are called boundary blow-up solutions of ∆u = u p . We show that a boundary blow-up solution exists in any bounded domain if 1 < p (Formula presented). In particular, when n = 2, there exists a boundary blow-up solution to ∆u = u p for all p ∈ (1, ∞). We also prove the uniqueness under the additional assumption that the domain satisfies the condition ∂Ω = ∂Ω.

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