DOI: 10.1515/anona-2023-0134 ISSN: 2191-950X
Infinitely many solutions for Hamiltonian system with critical growth
Yuxia Guo, Yichen Hu Abstract
In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain:
−
Δ
u
=
K
1
(
∣
y
∣
)
∣
v
∣
p
−
1
v
,
in
B
1
(
0
)
,
−
Δ
v
=
K
2
(
∣
y
∣
)
∣
u
∣
q
−
1
u
,
in
B
1
(
0
)
,
u
=
v
=
0
on
∂
B
1
(
0
)
,
\left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right.
where
K
1
(
r
)
{K}_{1}\left(r)
and
K
2
(
r
)
{K}_{2}\left(r)
are positive bounded functions defined in
[
0
,
1
]
\left[0,1]
,
B
1
(
0
)
{B}_{1}\left(0)
is the unit ball in
R
N
{{\mathbb{R}}}^{N}
, and
(
p
,
q
)
\left(p,q)
is a pair of positive numbers lying on the critical hyperbola
1
p
+
1
+
1
q
+
1
=
N
−
2
N
.
\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.
Under some suitable further assumptions on the functions