Abstract
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.