Boundary Behavior Of A Class Of Nonlinear Elliptic Equations Boundary Blow-up Of The Critical Situation

In this article,we consider the boundary behavior of solutions to the boundary blow-up elliptic problem△u=b(x)f(u),u≥0,x∈Ω,u|(?)Ω=∞(1.1) where the last condition means that u(x)→∞when d(x)=dist(x,(?)Ω)→0,Ω is a bounded domain with smooth boundary in RN,f satisfies(F1) f∈C[0,∞)∩C1(0,∞),f(0)=0and f(s)is increasing on(0,∞)；(F2)the Keller-Osserman([11],[15])condition(F3)there exist two functions f1∈C1[S0,∞)for some large S0>0and f2such that f(s):=f1(s)+f2(s), s≥S0；(F4) f’1(s)s/f1(s):=1+g(s),s≥S0, with g∈C1[S0,∞)satisfying g(s)>0, s≥S0,lim g(s)=0, s→∞(F5)either there exists a constant E1≠0such that or and there exists a constantμ<1such thatthe function b satisfies(B1) b∈Cα(Ω), is non-negative in Ω and positive near (?)Ω.(B2) there exists κ∈Λ and a positive constant b0such that: where Λ denotes the set of all positive non-decreasing functions in C1(O,ζ0)∩L1(Oζ0), which satisfy:Then for any solutions u of problem (1.1),Theorem1.1Let f satisfies (Fl)-(F5). Ifb satisfies (B1)-(B2), then for any solution u of problem (1.1) where ζ0-1/2-E2-(1-Dκ)(1/2+Cg), and ψ is the unique solution of the problem...