A second-order estimate for blow-up solutions of elliptic equations |
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Authors: | Shuibo Huang Qiaoyu Tian Shengzhi Zhang Jinhua Xi |
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Affiliation: | Department of Mathematics, Gansu Normal University for Nationalities, Hezuo Gansu, 747000, PR China |
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Abstract: | We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂RN. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is limu→∞f(ξu)/f(u)=ξρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory. |
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Keywords: | Boundary blow-up solutions Second-term asymptotic behavior Karamata regular variation theory |
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