Let
Ω
⊂
R
N
\Omega \subset \mathbb {R}^N
,
N
≥
1
N \geq 1
, be a bounded smooth connected open set and
a
:
Ω
×
R
N
→
R
N
\mathbf {a} : \Omega \times \mathbb {R}^N \to \mathbb {R}^N
be a map satisfying the hypotheses (H1)-(H4) below. Let
f
1
,
f
2
∈
L
l
o
c
1
(
Ω
)
f_1,f_2 \in \mathrm {L}_{loc}^{1} (\Omega )
with
f
2
≥
f
1
f_2 \geq f_1
,
f
1
≢
f
2
f_1 \not \equiv f_2
in
Ω
\Omega
and
u
1
,
u
2
∈
C
1
,
θ
(
Ω
¯
)
u_1, u_2 \in \mathcal {C}^{1,\theta } (\overline \Omega )
with
θ
∈
(
0
,
1
]
\theta \in (0,1]
be two weak solutions of
\[
(
P
i
)
−
d
i
v
(
a
(
x
,
∇
u
i
)
)
=
f
i
i
n
Ω
,
i
=
1
,
2.
(P_i)\quad -\mathrm {div} (\mathbf {a}(x,\nabla u_i)) = f_i \quad \mathrm {in }\,\Omega , \,\quad i=1,2.
\]
Suppose that
u
2
≥
u
1
u_2 \geq u_1
in
Ω
\Omega
. Then we show that
u
2
>
u
1
u_2 > u_1
in
Ω
\Omega
under the following assumptions: either
u
2
>
u
1
u_2>u_1
on
∂
Ω
\partial \Omega
, or
u
1
=
u
2
=
0
u_1=u_2=0
on
∂
Ω
\partial \Omega
and
u
1
≥
0
u_1 \geq 0
in
Ω
\Omega
. We also show a measure-theoretic version of the Strong Comparison Principle.