Affiliation:
1. Department of Mathematics , Faculty of Sciences , Ibn Tofail University , Kenitra , MOROCCO
Abstract
Abstract
In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem
{
−
Δ
u
=
|
u
|
2
∗
(
t
)
−
2
u
|
y
|
t
+
μ
|
u
|
q
−
2
u
in
Ω
,
u
=
0
on
∂
Ω
,
\begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u & \text { in } \Omega, \\ u=0 & \text { on } \partial \Omega,\end{cases}
where Ω is an open bounded domain in ℝ
N
, which contains some points (0,z*),
μ
>
0
,
1
<
q
<
2
,
2
∗
(
t
)
=
2
(
N
−
t
)
N
−
2
\mu>0,1<q<2,2^*(t)=\frac{2(N-t)}{N-2}
, 0 ≤ t < 2, x = (y, z) ∈ ℝ
k
× ℝ
N−k
, 2 ≤ k ≤ N. We prove that if
N
>
2
q
+
1
q
−
1
+
t
$N > 2{{q + 1} \over {q - 1}} + t$
, then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.