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.
Bibliographical noteFunding Information:
The author was partially supported by NRF Grant No. NRF-20151009350.
Received March 13, 2018; Accepted May 29, 2018. 2010 Mathematics Subject Classification. Primary 35J65; Secondary 35B05. Key words and phrases. blow-up, semi-linear equation, existence, uniqueness. The author was partially supported by NRF Grant No. NRF-20151009350.
Acknowledgment. This paper is based on a presentation by the author at the 2001 AMS sectional meeting, Williamstown, MA, and was supported in part by NSF Grant No. DMS-9971052.
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