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.
Nodal solutions for a critical Kirchhoff type problem in $\mathbb{R}^N$
