The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems

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Abstract

By constructing the comparison functions and the perturbed method, it is showed that any solution uC2(Ω) to the semilinear elliptic problems Δu=k(x)g(u), xΩ, u|Ω=+ satisfies limd(x)0u(x)Z(dμ(x))=[(2+σ)(2+ρ+σ)2c0(2+ρ)]1/ρ, where Ω is a bounded domain with smooth boundary in RN; limd(x)0k(x)dσ(x)=c0, 2<σ, c0>0, μ=2+σ2; gC1[0,), g0 and g(s)s is increasing on (0,), there exists ρ>0 such that limsg(sξ)g(s)=ξρ, ξ>0, Z(s)dt2G(t)=s, G(t)=0tg(s)ds.

Keywords

Semilinear elliptic equations
Large solutions
Precise asymptotic behaviour
Uniqueness

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This work is supported in part by National Natural Science Foundation of People's Republic of China under Grant numbers 10071066, 10251002.