Author:
Tian Guaiqi,An Yucheng,Suo Hongmin
Abstract
AbstractIn this work, we study the following Schrödinger-Poisson system $$ \textstyle\begin{cases} -\Delta _{H}u+\mu \phi u=\lambda u^{-\gamma}, &\text{in } \Omega , \\ -\Delta _{H}\phi =u^{2}, &\text{in } \Omega , \\ u>0, &\text{in } \Omega , \\ u=\phi =0, &\text{on } \partial \Omega , \end{cases} $$
{
−
Δ
H
u
+
μ
ϕ
u
=
λ
u
−
γ
,
in
Ω
,
−
Δ
H
ϕ
=
u
2
,
in
Ω
,
u
>
0
,
in
Ω
,
u
=
ϕ
=
0
,
on
∂
Ω
,
where $\Delta _{H}$
Δ
H
is the Kohn-Laplacian on the first Heisenberg group $\mathbb{H}^{1}$
H
1
, and $\Omega \subset \mathbb{H}^{1}$
Ω
⊂
H
1
is a smooth bounded domain, $\mu =\pm 1$
μ
=
±
1
, $0<\gamma <1$
0
<
γ
<
1
, and $\lambda >0$
λ
>
0
are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for $\mu =1$
μ
=
1
and each $\lambda >0$
λ
>
0
. Multiple solutions of the system are also considered for $\mu =-1$
μ
=
−
1
and $\lambda >0$
λ
>
0
small enough using the critical point theory for nonsmooth functional.
Funder
The Science and Technology Project of Bijie
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis