Affiliation:
1. Department of Mathematics , National Institute of Technology Rourkela , Rourkela , 769008 Odisha , India
Abstract
Abstract
In this paper, we study the following elliptic problem involving the (
p
(
y
)
,
q
(
y
)
{p(y),q(y)}
)-Laplacian operator:
{
-
div
(
a
(
y
)
|
∇
v
|
p
(
y
)
-
2
∇
v
)
+
b
(
y
)
|
v
|
p
(
y
)
-
2
v
-
div
(
|
∇
v
|
q
(
y
)
-
2
∇
v
)
=
g
(
y
,
v
)
,
y
∈
Ω
,
v
=
0
on
∂
Ω
,
\left\{\begin{aligned} \displaystyle{}{-}\operatorname{div}(a(y)|\nabla v|^{p(%
y)-2}\nabla v)+b(y)|v|^{p(y)-2}v-\operatorname{div}(|\nabla v|^{q(y)-2}\nabla v%
)&\displaystyle=g(y,v),&&\displaystyle y\in\Omega,\\
\displaystyle v&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial%
\Omega,\end{aligned}\right.
with Dirichlet boundary condition in an exterior domain Ω
(
⊂
ℝ
n
)
{(\subset\mathbb{R}^{n})}
with smooth boundary, where
1
<
q
(
y
)
<
p
(
y
)
<
n
1<q(y)<p(y)<n
. We prove the existence of solutions in
W
0
1
,
p
(
y
)
(
Ω
)
{W^{1,p(y)}_{0}(\Omega)}
for the superlinear case by using the Mountain Pass Theorem.
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p
&
q
p\&q
-problems with potentials vanishing at infinity,
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