Multiple solutions for p-Laplacian type problems with asymptotically p-linear terms via a cohomological index theory
Abstract: The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem \[ \left\{ \begin{array}{lr} - \divg (A(x,u)\, |\nabla u|^{p-2}\, \nabla u) \dfrac1p\, A_t(x,u)\, |\nabla u|^p\ =\ f(x,u) & \hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array}\right.\]\ud where $\Omega \subset \R^N$ is a bounded domain, $N\ge 2$,\ud $p > 1$, $A$ is a given function which admits partial derivative $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$\ud and $f$ is a… Show more
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The aim of this paper is investigating the existence of one or more
weak solutions of the coupled quasilinear elliptic system of gradient type
\textup{(P)}
{
-
div
(
A
(
x
,
u
)
|
∇
u
|
p
1
-
2
∇
u
)
+
1
p
1
A
u
(
x
,
u
)
|
∇
u
|
p
1
=
G
u
(
x
,
u
,
v
)
in
Ω
,
-
div
(
B
(
x
,
v
)
|
∇
v
|
p
2
-
2
∇
v
)
+
1
p
2
B
v
(
x
,
v
)
|
∇
v
|
p
2
=
G
v
(
x
,
u
,
v
)
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
\left\{\begin{aligned} \displaystyle-\operatorname{div}(A(x,u)|\nabla u|^{p_{1%
}-2}\nabla u)+\frac{1}{p_{1}}A_{u}(x,u)|\nabla u|^{p_{1}}&\displaystyle=G_{u}(%
x,u,v)&&\displaystyle\phantom{}\text{in~{}${\Omega}$,}\\
\displaystyle-\operatorname{div}(B(x,v)|\nabla v|^{p_{2}-2}\nabla v)+\frac{1}{%
p_{2}}B_{v}(x,v)|\nabla v|^{p_{2}}&\displaystyle=G_{v}(x,u,v)&&\displaystyle%
\phantom{}\text{in~{}${\Omega}$,}\\
\displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\partial%
\Omega}$,}\end{aligned}\right.
where
Ω
⊂
ℝ
N
{\Omega\subset\mathbb{R}^{N}}
is an open bounded domain,
p
1
{p_{1}}
,
p
2
>
1
{p_{2}>1}
and
A
(
x
,
u
)
{A(x,u)}
,
B
(
x
,
v
)
{B(x,v)}
are
𝒞
1
{\mathcal{C}^{1}}
-Carathéodory functions on
Ω
×
ℝ
{\Omega\times\mathbb{R}}
with partial derivatives
A
u
(
x
,
u
)
{A_{u}(x,u)}
, respectively
B
v
(
x
,
v
)
{B_{v}(x,v)}
, while
G
u
(
x
,
u
,
v
)
{G_{u}(x,u,v)}
,
G
v
(
x
,
u
,
v
)
{G_{v}(x,u,v)}
are given Carathéodory maps defined on
Ω
×
ℝ
×
ℝ
{\Omega\times\mathbb{R}\times\mathbb{R}}
which are partial derivatives of a function
G
(
x
,
u
,
v
)
{G(x,u,v)}
.
We prove that, even if the coefficients make the variational approach more difficult,
under suitable hypotheses functional
𝒥
{{\mathcal{J}}}
, related to problem (P),
admits at least one critical point in the “right” Banach space
The aim of this paper is investigating the existence of one or more
weak solutions of the coupled quasilinear elliptic system of gradient type
\textup{(P)}
{
-
div
(
A
(
x
,
u
)
|
∇
u
|
p
1
-
2
∇
u
)
+
1
p
1
A
u
(
x
,
u
)
|
∇
u
|
p
1
=
G
u
(
x
,
u
,
v
)
in
Ω
,
-
div
(
B
(
x
,
v
)
|
∇
v
|
p
2
-
2
∇
v
)
+
1
p
2
B
v
(
x
,
v
)
|
∇
v
|
p
2
=
G
v
(
x
,
u
,
v
)
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
\left\{\begin{aligned} \displaystyle-\operatorname{div}(A(x,u)|\nabla u|^{p_{1%
}-2}\nabla u)+\frac{1}{p_{1}}A_{u}(x,u)|\nabla u|^{p_{1}}&\displaystyle=G_{u}(%
x,u,v)&&\displaystyle\phantom{}\text{in~{}${\Omega}$,}\\
\displaystyle-\operatorname{div}(B(x,v)|\nabla v|^{p_{2}-2}\nabla v)+\frac{1}{%
p_{2}}B_{v}(x,v)|\nabla v|^{p_{2}}&\displaystyle=G_{v}(x,u,v)&&\displaystyle%
\phantom{}\text{in~{}${\Omega}$,}\\
\displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\partial%
\Omega}$,}\end{aligned}\right.
where
Ω
⊂
ℝ
N
{\Omega\subset\mathbb{R}^{N}}
is an open bounded domain,
p
1
{p_{1}}
,
p
2
>
1
{p_{2}>1}
and
A
(
x
,
u
)
{A(x,u)}
,
B
(
x
,
v
)
{B(x,v)}
are
𝒞
1
{\mathcal{C}^{1}}
-Carathéodory functions on
Ω
×
ℝ
{\Omega\times\mathbb{R}}
with partial derivatives
A
u
(
x
,
u
)
{A_{u}(x,u)}
, respectively
B
v
(
x
,
v
)
{B_{v}(x,v)}
, while
G
u
(
x
,
u
,
v
)
{G_{u}(x,u,v)}
,
G
v
(
x
,
u
,
v
)
{G_{v}(x,u,v)}
are given Carathéodory maps defined on
Ω
×
ℝ
×
ℝ
{\Omega\times\mathbb{R}\times\mathbb{R}}
which are partial derivatives of a function
G
(
x
,
u
,
v
)
{G(x,u,v)}
.
We prove that, even if the coefficients make the variational approach more difficult,
under suitable hypotheses functional
𝒥
{{\mathcal{J}}}
, related to problem (P),
admits at least one critical point in the “right” Banach space
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