Tóm tắt
We study the singular semilinear elliptic equation
∆u + f(., u) = 0
in
\mathcal D'(Ω)
, where
Ω ⊂ ℝ^n
(
n ≥ 1
) is a bounded domain of class
C^{1,1}. f : Ω × (0, ∞) → [0, ∞)
is such that
f(., u) \in L^1(Ω)
for
u > 0
and
u → f(x, u)
is continuous and nonincreasing for a.e.
x
in
Ω
. We assume that there exists a subset
Ω' ⊂ Ω
with positive measure such that
f(x, u) > 0
for
x\in Ω'
and
u > 0
and that
∫_Ω f(x, cd(x, ∂Ω))dx < ∞
for all
c > 0
. Then we show that there exists a unique solution
u
in
W_0^{1,1}(Ω)
such that
∆u\in L^1(Ω)
,
u > 0
a.e. in
Ω
.
