Affiliation:
1. College of Mathematics, Changchun Normal University , Changchun , 130032 , P.R. China
Abstract
Abstract
We deal with multiplicity of solutions to the following Schrödinger-Poisson-type system in this article:
Δ
H
u
−
μ
1
ϕ
1
u
=
∣
u
∣
2
u
+
F
u
(
ξ
,
u
,
v
)
,
in
Ω
,
−
Δ
H
v
+
μ
2
ϕ
2
v
=
∣
v
∣
2
v
+
F
v
(
ξ
,
u
,
v
)
,
in
Ω
,
−
Δ
H
ϕ
1
=
u
2
,
−
Δ
H
ϕ
2
=
v
2
,
in
Ω
,
ϕ
1
=
ϕ
2
=
u
=
v
=
0
,
on
∂
Ω
,
\left\{\begin{array}{ll}{\Delta }_{H}u-{\mu }_{1}{\phi }_{1}u={| u| }^{2}u+{F}_{u}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}v+{\mu }_{2}{\phi }_{2}v={| v| }^{2}v+{F}_{v}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}{\phi }_{1}={u}^{2},\hspace{1.0em}-{\Delta }_{H}{\phi }_{2}={v}^{2},\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {\phi }_{1}={\phi }_{2}=u=v=0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right.
where
Δ
H
{\Delta }_{H}
is the Kohn-Laplacian and
Ω
\Omega
is a smooth bounded region on the first Heisenberg group
H
1
{{\mathbb{H}}}^{1}
,
μ
1
{\mu }_{1}
, and
μ
2
{\mu }_{2}
are some real parameters, and
F
=
F
(
x
,
u
,
v
)
,
F
u
=
∂
F
∂
u
F=F\left(x,u,v),{F}_{u}=\frac{\partial F}{\partial u}
,
F
v
=
∂
F
∂
u
{F}_{v}=\frac{\partial F}{\partial u}
satisfying natural growth conditions. By the limit index theory and the concentration compactness principles, we prove that the aforementioned system has multiplicity of solutions for
μ
1
,
μ
2
<
∣
Ω
∣
−
1
2
S
{\mu }_{1},{\mu }_{2}\lt {| \Omega | }^{-\tfrac{1}{2}}S
, where
S
S
is the best Sobolev constant. The novelties of this article are the presence of critical nonlinear term, and the system is set on the Heisenberg group.