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Asymptotic behavior of positive solutions of a semilinear Dirichlet problem. (English) Zbl 1213.31006

Summary: We study the asymptotic behavior of the classical unique positive solution to the following semilinear boundary value problem

Δu+a(x)u α =0,xΩ,u>0inΩ,u| Ω =0·

Here Ω is a bounded C 1,1 domain, α<1, and the function a is in C loc γ (Ω), 0<γ<1, such that there exists c>0 satisfying for each xΩ,

1 ca(x)δ(x) λ exp- δ(x) η z(t) t d tc,

where λ2, η>d=diamΩ, δ(x)=dist(x,Ω), and z is a continuous function on [0,η] with z(0)=0.

MSC:
31C15Generalizations of potentials and capacities
34B27Green functions
35K10Second order parabolic equations, general
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