Abstract
AbstractIn this paper, we consider the existence of solutions of the following Kirchhoff-type problem: $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$
{
−
(
a
+
b
∫
R
3
|
∇
u
|
2
d
x
)
Δ
u
+
V
(
x
)
u
=
f
(
x
,
u
)
,
in
R
3
,
u
∈
H
1
(
R
3
)
,
where $a,b>0$
a
,
b
>
0
are constants, and the potential $V(x)$
V
(
x
)
is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.
Funder
Doctoral research project of Guizhou Normal University
National Natural Science Foundation of Chin
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
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