Remarks on large solutions of a class of semilinear elliptic equations

In this paper, we show existence, uniqueness and exact asymptotic behavior of solutions near the boundary to a class of semilinear elliptic equations − Δ u = λ g ( u ) − b ( x ) f ( u ) in Ω, where λ is a real number, b ( x ) > 0 in Ω and vanishes on ∂Ω. The special feature is to consider g ( u ) and f ( u ) to be regularly varying at infinity and b ( x ) is vanishing on the boundary with a more general rate function. The vanishing rate of b ( x ) determines the exact blow-up rate of the large solutions. And the exact blow-up rate allows us to obtain the uniqueness result.