Author:
Brandolini Barbara,Cîrstea Florica C.
Abstract
AbstractWe prove the existence of weak solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$\begin{aligned}{\mathcal {A}} u+\Phi (x,u,\nabla u)=\Psi (u,\nabla u)+\mathfrak Bu +f \end{aligned}$$
A
u
+
Φ
(
x
,
u
,
∇
u
)
=
Ψ
(
u
,
∇
u
)
+
B
u
+
f
on a bounded open subset $$\Omega \subset {\mathbb {R}}^N$$
Ω
⊂
R
N
$$(N\ge 2)$$
(
N
≥
2
)
, where $$f\in L^1(\Omega )$$
f
∈
L
1
(
Ω
)
is arbitrary. Our models are $$ \mathcal Au=-\sum _{j=1}^N \partial _j (|\partial _j u|^{p_j-2}\partial _j u)$$
A
u
=
-
∑
j
=
1
N
∂
j
(
|
∂
j
u
|
p
j
-
2
∂
j
u
)
and $$\Phi (u,\nabla u)=\left( 1+\sum _{j=1}^N {\mathfrak {a}}_j |\partial _j u|^{p_j}\right) |u|^{m-2}u$$
Φ
(
u
,
∇
u
)
=
1
+
∑
j
=
1
N
a
j
|
∂
j
u
|
p
j
|
u
|
m
-
2
u
, with $$m,p_j>1$$
m
,
p
j
>
1
,$${\mathfrak {a}}_j\ge 0$$
a
j
≥
0
for $$1\le j\le N$$
1
≤
j
≤
N
and $$\sum _{k=1}^N (1/p_k)>1$$
∑
k
=
1
N
(
1
/
p
k
)
>
1
. The main novelty is the inclusion of a possibly singular gradient-dependent term $$\Psi (u,\nabla u)=\sum _{j=1}^N |u|^{\theta _j-2}u\, |\partial _j u|^{q_j}$$
Ψ
(
u
,
∇
u
)
=
∑
j
=
1
N
|
u
|
θ
j
-
2
u
|
∂
j
u
|
q
j
, where $$\theta _j>0$$
θ
j
>
0
and $$0\le q_j<p_j$$
0
≤
q
j
<
p
j
for $$1\le j\le N$$
1
≤
j
≤
N
. Under suitable conditions, we prove the existence of solutions by distinguishing two cases: 1) for every $$1\le j\le N$$
1
≤
j
≤
N
, we have $$\theta _j> 1$$
θ
j
>
1
and 2) there exists $$1\le j\le N$$
1
≤
j
≤
N
such that $$\theta _j\le 1$$
θ
j
≤
1
. In the latter situation, assuming that $$f \ge 0$$
f
≥
0
a.e. in $$\Omega $$
Ω
, we obtain non-negative solutions for our problem.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
Cited by
1 articles.
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