Bounded very weak solutions for some non-uniformly elliptic equation with L^1 datum
Abstract: Abstract. In this paper we obtain the existence of bounded very weak solutions for the Dirichlet boundary value problem of a class of non-uniformly elliptic equations with L 1 integrability conditions by using the regularizing effect of the interaction between the coefficient of lower order term and the datum in the right-hand side.
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We study of the regularizing effect of the interaction between
the coefficient of the zero-order term and the lower-order term in quasilinear Dirichlet problems whose model is
∫
Ω
M
(
x
,
u
)
∇
u
⋅
∇
φ
+
∫
Ω
a
(
x
)
u
φ
=
∫
Ω
b
(
x
)
|
∇
u
|
q
φ
+
∫
Ω
f
(
x
)
φ
for all
φ
∈
W
0
1
,
2
(
Ω
)
∩
L
∞
(
Ω
)
,
\int_{\Omega}M(x,u)\nabla u\cdot\nabla\varphi+\int_{\Omega}a(x)u\varphi=\int_{%
\Omega}b(x)|\nabla u|^{q}\varphi+\int_{\Omega}f(x)\varphi\quad\text{for all }%
\varphi\in W_{0}^{1,2}(\Omega)\cap L^{\infty}(\Omega),
where Ω is a bounded open set of
ℝ
N
{\mathbb{R}^{N}}
,
M
(
x
,
s
)
{M(x,s)}
is a Carathéodory matrix on
Ω
×
ℝ
{\Omega\times\mathbb{R}}
which is elliptic (that is,
M
(
x
,
s
)
ξ
⋅
ξ
≥
α
|
ξ
|
2
>
0
{M(x,s)\xi\cdot\xi\geq\alpha|\xi|^{2}>0}
for every
(
x
,
s
,
ξ
)
∈
Ω
×
ℝ
×
(
ℝ
N
∖
{
0
}
)
{(x,s,\xi)\in\Omega\times\mathbb{R}\times(\mathbb{R}^{N}\setminus\{0\})}
) and bounded
(that is,
|
M
(
x
,
s
)
|
≤
β
{|M(x,s)|\leq\beta}
for every
(
x
,
s
)
∈
Ω
×
ℝ
{(x,s)\in\Omega\times\mathbb{R}}
),
b
(
x
)
∈
L
2
2
-
q
(
Ω
)
{b(x)\in L^{\frac{2}{2-q}}(\Omega)}
,
1
<
q
<
2
{1<q<2}
and
0
≤
a
(
x
)
∈
L
1
(
Ω
)
{0\leq a(x)\in L^{1}(\Omega)}
.
We prove the existence of a weak solution u belonging to
W
0
1
,
2
(
Ω
)
{W_{0}^{1,2}(\Omega)}
and to
L
∞
(
Ω
)
{L^{\infty}(\Omega)}
when either
b
∈
L
2
m
2
-
q
(
Ω
)
for some
m
>
N
2
and
\displaystyle b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m>\frac{N}{2}%
\text{ and}
(0.1)
∃
Q
>
0
such that
|
f
(
x
)
|
≤
Q
a
(
x
)
We study of the regularizing effect of the interaction between
the coefficient of the zero-order term and the lower-order term in quasilinear Dirichlet problems whose model is
∫
Ω
M
(
x
,
u
)
∇
u
⋅
∇
φ
+
∫
Ω
a
(
x
)
u
φ
=
∫
Ω
b
(
x
)
|
∇
u
|
q
φ
+
∫
Ω
f
(
x
)
φ
for all
φ
∈
W
0
1
,
2
(
Ω
)
∩
L
∞
(
Ω
)
,
\int_{\Omega}M(x,u)\nabla u\cdot\nabla\varphi+\int_{\Omega}a(x)u\varphi=\int_{%
\Omega}b(x)|\nabla u|^{q}\varphi+\int_{\Omega}f(x)\varphi\quad\text{for all }%
\varphi\in W_{0}^{1,2}(\Omega)\cap L^{\infty}(\Omega),
where Ω is a bounded open set of
ℝ
N
{\mathbb{R}^{N}}
,
M
(
x
,
s
)
{M(x,s)}
is a Carathéodory matrix on
Ω
×
ℝ
{\Omega\times\mathbb{R}}
which is elliptic (that is,
M
(
x
,
s
)
ξ
⋅
ξ
≥
α
|
ξ
|
2
>
0
{M(x,s)\xi\cdot\xi\geq\alpha|\xi|^{2}>0}
for every
(
x
,
s
,
ξ
)
∈
Ω
×
ℝ
×
(
ℝ
N
∖
{
0
}
)
{(x,s,\xi)\in\Omega\times\mathbb{R}\times(\mathbb{R}^{N}\setminus\{0\})}
) and bounded
(that is,
|
M
(
x
,
s
)
|
≤
β
{|M(x,s)|\leq\beta}
for every
(
x
,
s
)
∈
Ω
×
ℝ
{(x,s)\in\Omega\times\mathbb{R}}
),
b
(
x
)
∈
L
2
2
-
q
(
Ω
)
{b(x)\in L^{\frac{2}{2-q}}(\Omega)}
,
1
<
q
<
2
{1<q<2}
and
0
≤
a
(
x
)
∈
L
1
(
Ω
)
{0\leq a(x)\in L^{1}(\Omega)}
.
We prove the existence of a weak solution u belonging to
W
0
1
,
2
(
Ω
)
{W_{0}^{1,2}(\Omega)}
and to
L
∞
(
Ω
)
{L^{\infty}(\Omega)}
when either
b
∈
L
2
m
2
-
q
(
Ω
)
for some
m
>
N
2
and
\displaystyle b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m>\frac{N}{2}%
\text{ and}
(0.1)
∃
Q
>
0
such that
|
f
(
x
)
|
≤
Q
a
(
x
)
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