Abstract
Abstract
The purpose of this article is to state some weighted Sobolev embedding involving functions that vanishing only in a direction. In this setting we prove a weighted Trudinger–Moser type inequality and as an application, we addressed the existence of solutions to a class of elliptic equation of the form
−
div
(
a
(
x
)
∇
u
)
+
V
(
x
)
u
=
K
(
x
)
f
(
u
)
in
R
2
,
where the nonlinearity f has exponential critical growth in sense of Trudinger–Moser.