DOI: 10.1515/dema-2025-0293 ISSN: 2391-4661
Ground state sign-changing solution for generalized quasilinear Kirchhoff–Schrödinger–Poisson system
Yuanyuan Yang, Jing Zhang Abstract
This paper explores the existence of ground state sign-changing solutions of the following generalized quasilinear Kirchhoff–Schrödinger–Poisson system
1
+
b
∫
R
3
g
2
(
u
)
|
∇
u
|
2
d
x
[
−
d
i
v
(
g
2
(
u
)
∇
u
)
+
g
(
u
)
g
′
(
u
)
|
∇
u
|
2
]
+
V
(
x
)
u
+
ϕ
G
(
u
)
g
(
u
)
=
K
(
x
)
f
(
u
)
,
x
∈
R
3
,
−
Δ
ϕ
=
G
2
(
u
)
,
x
∈
R
3
,
$$\begin{cases}\left(1+b{\int }_{ {\mathbb{R}}^{3}}{g}^{2}\left(u\right)\vert \nabla u{\vert }^{2}\mathrm{d}x\right)\left[ - \mathrm{d}\mathrm{i}\mathrm{v}\left({g}^{2}\left(u\right)\nabla u\right)+g\left(u\right){g}^{\prime }\left(u\right)\vert \nabla u{\vert }^{2}\right]\hfill \\ +V\left(x\right)u+\phi G\left(u\right)g\left(u\right)=K\left(x\right)f\left(u\right),\hfill & x\in {\mathbb{R}}^{3},\hfill \\ -{\Delta}\phi ={G}^{2}\left(u\right),\hfill & x\in {\mathbb{R}}^{3},\hfill \end{cases}$$
where
b
> 0,
V
(
x
) and
K
(
x
) are positive functions,
f
is a continuous function,
g
∈
C
1
(
R
,
R
+
)
$g\in {\mathcal{C}}^{1}\left(\mathbb{R}, {\mathbb{R}}^{+}\right)$
and
G
(
u
)
=
∫
0
u
g
(
s
)
d
s
$G\left(u\right)={\int }_{0}^{u}g\left(s\right)\mathrm{d}s$
. We employ the generalized Nehari manifold method combined with advanced variational techniques to demonstrate the existence of ground state sign-changing solutions
v
b
. Furthermore, we establish that the energy of
v
b
is strictly larger than twice that of the ground state solutions of the Nehari type. The convergence behavior of the solutions is also analyzed as
b
→ 0.