Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 13, pp. 1–15.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR

SCHRÖDINGER-KIRCHHOFF EQUATIONS

WITH INDEFINITE POTENTIALS

SHUAI JIANG, LI-FENG YIN

Abstract. We consider a class of Schrödinger-Kirchhoff equations in R3 with
a general nonlinearity g and coercive sign-changing potential V so that the

Schrödinger operator −a∆ + V is indefinite. The nonlinearity considered here

satisfies the Ambrosetti-Rabinowitz type condition g(t)t ≥ µG(t) > 0 with
µ > 3. We obtain the existence of nontrivial solutions for this problem via

Morse theory.

1. Introduction

In this article, we consider the Schrödinger-Kirchhoff type problem

−
(
a+ b

∫
R3

|∇u|2dx
)

∆u+ V (x)u = g(u), u ∈ H1(R3), (1.1)

where a and b are positive constants, V ∈ C(R3) is the potential and g ∈ C(R) is
the nonlinearity.

The above problem is nonlocal as the term
∫
R3 |∇u|2dx implies that (1.1) is

not a pointwise identity. This feature causes some mathematical difficulties, which
make the investigation of (1.1) particularly interesting. Problem (1.1) arises in
an interesting physical context. Indeed, if we set V (x) = 0 and replace R3 by a
bounded domain Ω ⊂ R3, problem (1.1) reduces to the Dirichlet problem

−
(
a+ b

∫
Ω

|∇u|2dx
)

∆u = g(u) in Ω,

u = 0 on ∂Ω,
(1.2)

which is related to the stationary analogue of the equation

ρ
∂2u

∂t2
−
(P0

h
+

E

2L

∫ L

0

|∂u
∂x
|2dx

)∂2u

∂x2
= 0, (1.3)

proposed by Kirchhoff [10] as an extension of the classical D’Alembert’s wave equa-
tion for free vibrations of elastic strings, where ρ, P0, h, E and L are positive

2020 Mathematics Subject Classification. 35B38, 35J20, 35J60.
Key words and phrases. Schrödinger-Kirchhoff equations; Palais-Smale condition;

Morse theory.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted March 28, 2022. Published February 10, 2023.

1



2 S. JIANG, L.-F. YIN EJDE-2023/13

constants. Lions [15] introduced an abstract functional analysis framework for the
equation

utt −
(
a+ b

∫
Ω

|∇u|2dx
)

∆u = g(x, u). (1.4)

Since then, problem (1.4) have received much attention.
Problem (1.1) has been studied extensively in recent years (in bounded or un-

bounded domain), see, for example, [1, 7, 14, 22, 23, 26, 28] and references therein.
We emphasize that in all these papers, the potential V is assumed to be non-
negative. In this case, the quadratic part of the variational functional Φ given in
(2.1) is positively definite, the zero function u = 0 is a local minimizer of Φ and the
mountain pass theorem [2] can be applied. However, when the potential V is nega-
tive somewhere so that the quadratic part of Φ is indefinite, the zero function u = 0
is no longer a local minimizer of Φ, the mountain pass theorem is not applicable
anymore. For the semilinear problem (b = 0)

−∆u+ V (x)u = g(u)

with indefinite Schrödinger operator −∆ + V , one usually applies the linking theo-
rem to obtain a solution, see e.g. [11, 19]. For problem (1.1), it seems hard to verify
the linking geometry because of the nonlocal term

∫
R3 |∇u|2dx, which prevents the

functional Φ to be nonpositive on the negative space of the Schrödinger operator,
see Remark 1.2 for detail. Hence, the classical linking theorem [25, Lemma 2.12] is
also not applicable.

For this reason, there are very few results about (1.1) with indefinite potential.
To the best of our knowledge, the first work on this situation is due to Chen and
Liu [6]. To overcome the above difficulty, a crucial observation in [6] is that Φ has
a local linking at u = 0, thus for certain i ∈ N, the i-th critical group of Φ at u = 0
is nontrivial. Assuming that g is 3-superlinear

lim
|t|→∞

g(t)

t3
= +∞, (1.5)

with suitable technical conditions it was shown in [6] that all critical groups of Φ
at infinity are trivial, then a nonzero critical point of Φ can be found via Morse
theory. See also [13] for a recent result for 3-superlinear nonlinearity.

For other results on the indefinite problem (1.1), we mention [27] and [9], where
the nonlinearity g is sublinear and subquadratic, respectively. In these two papers
the variational functional is coercive. The three critical points theorem of Liu and
Su [16] and the classical Clark theorem are applied to obtain multiple solutions.

Having the above results in mind, it is natural to ask what will happen when g
is 2-superlinear (g satisfies (1.5) with 3 replaced by 2)? This is the motivation of
the current paper.

The main difficulty under this assumption is that it is not know whether Palais-
Smale (or even Cerami) sequences are bounded or not. Motivated by Liu and
Mosconi [20] on the study of nonlinear Schrödinger-Poisson systems (see also [8]),

we add a dummy variable and consider an augmented functional Φ̃ : R ×X → R,
see (2.4). It turns out that Φ̃ satisfies the (PS) condition (see Lemma 2.6), and

if (s̄, ū) is a critical point of Φ̃, then ū is a critical point of Φ (see Lemma 2.5).

Moreover, Φ̃ has a local linking at zero, and all critical groups of Φ̃ at infinity are
trivial. Eventually, a nonzero critical point of Φ̃ can be obtained by using Morse
theory, which give rise to a nonzero critical point of Φ.



EJDE-2023/13 SCHRÖDINGER-KIRCHHOFF EQUATIONS 3

Without loss of generality, we assume that a = b = 1. Then the problem (1.1)
can be rewritten as follows

−
(

1 +

∫
R3

|∇u|2dx
)

∆u+ V (x)u = g(u), u ∈ H1(R3). (1.6)

Let H1(R3) be the standard Sobolev space. If V ∈ C(R3) is bounded from
below, we can choose a constant m > 2 such that

Ṽ (x) := V (x) +m ≥ m

2
≥ 1, for all x ∈ R3. (1.7)

On the linear subspace

X =
{
u ∈ H1(R3) :

∫
V (x)u2 <∞

}
, (1.8)

where from now on all integrals are taken over R3 except stated explicitly, we equip
with the inner product

(u, v) =

∫ (
∇u · ∇v + Ṽ (x)uv

)
and the corresponding norm ‖ · ‖. Note that if the assumptions (A1) below holds,
then X is a Hilbert space and by Bartsch and Wang [4] we have a compact embed-
ding X ↪→ Ls(R3) for s ∈ [2, 6).

Now we present our assumptions on the potential V (x) and the nonlinearity
g(u).

(A1) V ∈ C(R3) is bounded from below and |{V ≤ k}| <∞ for all k ∈ R, where
| · | is the Lebesgue measure on R3.

(A2) V ∈ C1(R3) and there exists R > 0 such that V (x) − ∇V (x) · x ≥ 0 for
|x| ≥ R.

(A3) There exists κ > 0 such that

|∇V (x) · x| ≤ κ (V (x) +m) =: κṼ (x)

for all x ∈ R3, here m is the constant from (1.7).
(A4) g ∈ C(R) and there exist C > 0 and p ∈ (2, 6) such that

|g(t)| ≤ C
(
|t|+ |t|p−1

)
.

(A5) There exists µ > 3 such that g(t)t ≥ µG(t) > 0 for all t ∈ R \ {0}.
Consider the bilinear form

Q(u, v) =
1

2

∫
(∇u · ∇v + V (x)uv) u, v ∈ X, (1.9)

then X = X+ ⊕X− ⊕X0, where X+, X− and X0 are positive, negative, and null
eigenspaces of the Schrödinger operator. It is well known that there exist constants
η± > 0 such that

±Q(u, u) ≥ η±‖u±‖2, for u ∈ X± ⊕X0, respectively. (1.10)

We are now ready to state our main result.

Theorem 1.1. Suppose that (A1)–(A5) hold. If either

(1) dimX− > 0, dimX0 = 0, or

(2) dimX0 > 0 and G(t) =
∫ t

0
g(τ)dτ ≥ c|t|ν for some ν < 4,

then problem (1.6) has at least one nontrivial solution.



4 S. JIANG, L.-F. YIN EJDE-2023/13

Remark 1.2. As we have mentioned, because of the nonlocal term
∫
R3 |∇u|2dx,

our functional Φ (see (2.1)) does not satisfy the geometric assumption of the linking
theorem. For R > r > 0 set

N = {u ∈ X+ : ‖u‖ = r}, M = {u ∈ X− ⊕X0 ⊕ φR+ : ‖u‖ ≤ R},

where φ ∈ X+ \ {0}. Then we have

b = inf
N

Φ > 0, sup
u∈∂M,‖u‖=R

Φ < 0

provided R is large enough and r is small enough. However, because the integral(∫
|∇u|2

)2
in our functional, Φ may be very large, it is possible that Φ(u) > b for

some u ∈ ∂M ∩ (X− ⊕X0). Therefore, the following geometric assumption of the
linking theorem

inf
N

Φ > sup
∂M

Φ

can not be satisfied. Thus, the linking theorem is not applicable for problem (1.6).

2. Palais-Smale condition

Now, we investigate the functional Φ. Under the assumptions (A1) and (A4) we
can show that the functional Φ : X → R,

Φ(u) =
1

2

∫ (
|∇u|2 + V (x)u2

)
+

1

4

(∫
|∇u|2

)2

−
∫
G(u) (2.1)

is well defined and of class C1. The derivative of Φ is given by

〈DΦ(u), v〉 =
(

1 +

∫
|∇u|2

)∫
∇u · ∇v +

∫
V (x)uv −

∫
g(u)v.

for u, v ∈ X. Consequently, critical points of Φ are weak solutions of problem (1.6).

Proposition 2.1 ([20, Lemma 3.1]). Assume (A3) holds and let Ṽ := V +m. Then
for any t > 0 and x ∈ R3, we have

Ṽ (tx) ≤ max{tκ, t−κ}Ṽ (x). (2.2)

For t > 0 and u ∈ X we define

ut(x) = tu(
x

t
), (2.3)

and define on the Hilbert space R ×X (with natural norm ‖(s, u)‖2 = s2 + ‖u‖2)
the augmented functional

Φ̃(s, u) :=
s2

2
+ Φ(ues). (2.4)

Remark 2.2. Obviously, for s, t > 0, from (2.3) we have (ut)s = uts = (us)t.

Lemma 2.3. Suppose that (A3) and (A4) hold. Then the functional Φ̃ is well
defined on R×X, of class C1, and

Φ̃(s, u) =
s2

2
+
e3s

2

∫
|∇u|2 +

e6s

4

(∫
|∇u|2

)2

+
e5s

2

∫
V (xes)u2 − e3s

∫
G(esu),

(2.5)



EJDE-2023/13 SCHRÖDINGER-KIRCHHOFF EQUATIONS 5

〈∂uΦ̃(s, u), ϕ〉 = e3s

∫
∇u∇ϕ+ e6s

(∫
|∇u|2

)∫
∇u∇ϕ

+ e5s

∫
V (xes)uϕ− e4s

∫
g(esu)ϕ,

(2.6)

∂sΦ̃(s, u) = s+
3e3s

2

∫
|∇u|2 +

3e6s

2

(∫
|∇u|2

)2

+
e5s

2

∫
[5V (xes) +∇V (xes) · xes]u2

− e3s

∫
[3G(esu) + g(esu)esu] .

(2.7)

Proof. By changing variables, we obtain(∫
|∇ut|2

)2

= t6
(∫
|∇u|2

)2

≤ t6‖u‖4.

Similarly, ∫
|∇ut|2 = t3

∫
|∇u|2 ≤ t3‖u‖2,∣∣ ∫ G(ut)

∣∣ ≤ C|t|5 ∫ u2 + C|t|p+3

∫
|u|p.

By the continuity of V , changing variables x = yt on any fixed ball BR and applying
Proposition 2.1, we obtain∫

BR

Ṽ (x)u2
t = t5

∫
BR/t

Ṽ (tx)u2 ≤ t5 max{tκ, t−κ}
∫
BR/t

Ṽ (x)u2 ≤ ct‖u‖2.

Letting R→ +∞ we deduce that∫
Ṽ (x)u2

t ≤ ct‖u‖2.

From the above estimates and continuous embeddings,

Φ(ut) =
1

2

∫ (
|∇ut|2 + V (x)u2

t

)
+

1

4

(∫
|∇ut|2

)2

−
∫
G(ut)

≤ t3

2
‖u‖2 +

ct
2
‖u‖2 +

t6

4
‖u‖4 + C|t|5‖u‖2 + C|t|p+3‖u‖p.

Together with (2.4), it is easy to see that Φ̃ is well defined. Formula (2.5) directly
follows from changing variables.

Formula (2.6) can be computed in a standard way, while (2.7) is obtained by
differentiating under the integral sign in (2.5). By (A4), we deduce that

|3G(esu) + g(esu)esu| ≤ C(e2s|u|2 + eps|u|6). (2.8)

It follows from |V | ≤ Ṽ +m, (A3) and Proposition 2.1 that

|5V (xes) +∇V (xes) · xes| ≤ 5|V (xes)|+ |∇V (xes) · xes|

≤ 5m+ (5 + κ)Ṽ (xes)

≤ 5m+ (5 + κ)eκ|s|Ṽ (x).

(2.9)

Noting that, by the dominated convergence theorem, both

s 7→
∫
|5V (xes) +∇V (xes) · xes|, s 7→

∫
|3G(esu) + g(esu)esu|



6 S. JIANG, L.-F. YIN EJDE-2023/13

are continuous, standard arguments yields the differentiation formula (2.7). Finally,
estimates (2.8) and (2.9) ensure the continuity of the corresponding Nemitskii op-

erators appearing in (2.6) and (2.7), so that Φ̃ is of class C1. �

Lemma 2.4 (Pohožaev identity). Suppose that (A3) and (A4) hold. Let u ∈ X be
a weak solution to problem (1.6), then we have the following Pohožaev identity,

1

2

∫
|∇u|2 +

3

2

∫
V (x)u2 +

1

2

∫
(∇V (x), x)u2 +

1

2

(∫
|∇u|2

)2

− 3

∫
G(u) = 0.

(2.10)
Moreover,

d

dt

∣∣
t=1

Φ(ut) = 0.

Proof. The proof of Pohožaev identity is standard, so we omit it and refer the
reader to [12, Lemma 2.1]. Now, it suffices to show that

d

dt

∣∣
t=1

Φ(ut) = 0.

By changing variables and (2.1) , we have

Φ(ut) =
t3

2

∫
|∇u|2 +

t5

2

∫
V (xt)u2 +

t6

4

(∫
|∇u|2

)2

− t3
∫
G(tu). (2.11)

Then, by (2.10), (2.11), and DΦ(u) = 0, we obtain

dΦ(ut)

dt

∣∣
t=1

=
3

2

∫
|∇u|2 +

5

2

∫
V (x)u2 +

1

2

∫
(∇V (x), x)u2

+
3

2

(∫
|∇u|2

)2

−
∫

[3G(u) + g(u)u]

= 〈DΦ(u), u〉+
1

2

∫
|∇u|2 +

3

2

∫
V (x)u2

+
1

2

∫
(∇V (x), x)u2 +

1

2

(∫
|∇u|2

)2

− 3

∫
G(u) = 0. �

In the following, we denote by D̃Φ̃ the total differential of Φ̃ with respect to both
variables s and u.

Lemma 2.5. Suppose that (A3) and (A4) hold, then

D̃Φ̃(s̄, ū) = 0 ⇔ s̄ = 0 and DΦ(ū) = 0.

Proof. The proof is similar to the proof of [20, Lemma 3.5], we include it here for
the reader’s convenience.
(⇐) From the definition of D̃Φ̃, we have

D̃Φ̃(s̄, ū)[l, ϕ] = 〈∂uΦ̃(s̄, ū), ϕ〉+ ∂sΦ̃(s̄, ū)l, for all [l, ϕ] ∈ R×X. (2.12)

Note that DΦ(ū) = 0 implies

〈∂uΦ̃(0, ū), ϕ〉 = 〈DΦ(ū), ϕ〉 = 0.

Therefore, it suffices to prove that ∂sΦ̃(0, ū) = 0. From DΦ(ū) = 0, Lemma 2.4
gives

d

dt

∣∣
t=1

Φ(ūt) = 0. (2.13)



EJDE-2023/13 SCHRÖDINGER-KIRCHHOFF EQUATIONS 7

The map t 7→ Φ(ūt) is C1 by Lemma 2.3 and

Φ(ūt) = Φ̃(log t, ū)− log2 t

2
.

Now (2.13) gives

0 =
d

dt

∣∣
t=1

(
Φ̃(log t, ū)− log2 t

2

)
=
(
∂sΦ̃(log t, ū)

1

t
− log t

t

)∣∣
t=1

= ∂sΦ̃(0, ū).

(⇒) If D̃Φ̃(s̄, ū) = 0, combining (2.4), we immediately infer that

0 = ∂uΦ̃(s̄, ū) = ∂uΦ(ūes̄) = DΦ(ūes̄).

Thus, we only have to prove that s̄ = 0. Applying Lemma 2.4 to v := ūes̄ gives

d

dt

∣∣
t=1

Φ(vt) = 0.

The map t 7→ Φ(vt) is C1 by Lemma 2.3 and

Φ(vt) = Φ(ūtes̄) = Φ̃(s̄+ log t, ū)− (s̄+ log t)2

2
.

By the chain rule and ∂sΦ̃(s̄, ū) = 0 we have

0 =
d

dt

∣∣
t=1

Φ(vt)

=
(
∂sΦ̃(s̄+ log t, ū)

1

t
− (s̄+ log t)

t

)∣∣
t=1

= ∂sΦ̃(s̄, ū)− s̄ = −s̄. �

Lemma 2.6. Suppose that (A1)–(A5) hold. Then Φ̃ satisfies the (PS) condition.

Proof. Let {(sn, un)} be a (PS) sequence for Φ̃ in R×X, that is

sup
n
|Φ̃(sn, un)| <∞, D̃Φ̃(sn, un)→ 0.

Choose λ ∈ (3, µ), where µ > 3 is given by (A5). Then

(λ+ 3)Φ̃(s, u)− ∂sΦ̃(s, u)

=
λ+ 3

2
s2 − s+

λ

2
e3s

∫
|∇u|2 +

λ− 3

4
e6s
(∫
|∇u|2

)2

+
e5s

2

∫
[(λ− 2)V (xes)−∇V (xes) · xes]u2 + e3s

∫
[g(esu)esu− λG(esu)]

≥ λ+ 3

2
s2 − s+

λ

2
e3s

∫
|∇u|2 +

λ− 3

4
e6s
(∫
|∇u|2

)2

+
e5s

2

∫
[(λ− 2)V (xes)−∇V (xes) · xes]u2 + (µ− λ)e3s

∫
G(esu).

(2.14)
The third integral is bounded from below through (A1)–(A3), and Hölder’s inequal-
ity. Indeed, we set

v(x) = esu
( x
es

)
, Wλ(x) = (λ− 2)V (x)−∇V (x) · x.

Then, by changing variables,

e5s

∫
[(λ− 2)V (xes)−∇V (xes) · xes]u2 =

∫
Wλv

2. (2.15)



8 S. JIANG, L.-F. YIN EJDE-2023/13

As Wλ is bounded on bounded sets, we let Cλ ∈ R be such that Wλ ≥ −Cλ in BR,
where R is given in (A2), then∫

BR

Wλv
2 ≥ −Cλ

∫
BR

v2 ≥ −Cλ
(∫
|v|µ

)2/µ

. (2.16)

We write the integral over R3 \BR as the sum of the integrals over the intersections
of R3 \ BR with {V ≥ 0} and {V < 0}. Obviously, assumption (A1) implies that
{V < 0} has finite measure. Because λ− 2 > 1, assumption (A2) implies that

Wλ(x) = (λ− 2)V (x)−∇V (x) · x ≥ V (x)−∇V (x) · x ≥ 0

for x ∈ {V ≥ 0} \BR.
On the other hand, by (A3) and V ≥ −m we have

Wλ(x) ≥ (λ− 2)V − κ(V +m) ≥ − ((λ− 2) + κ)m on {V < 0}.

For some possibly larger Cλ, we have∫
R3\BR

Wλv
2 ≥

∫
{V <0}\BR

Wλv
2 ≥ −Cλ

∫
{V <0}

v2

≥ −Cλ|{V < 0}|(µ−2)/µ
(∫
|v|µ

)2/µ

.

(2.17)

Combining (2.15), (2.16), and (2.17), we obtain

e5s

∫
[(λ− 2)V (xes)−∇V (xes) · xes]u2 ≥ −Cλ

(∫
|v|µ

)2/µ

. (2.18)

Condition (A5) implies that for some C > 0, G(t) ≥ C|t|µ. We deduce

(µ− λ)e3s

∫
G(esu) = (µ− λ)

∫
G(v) ≥ (µ− λ)C

∫
|v|µ. (2.19)

From (2.14), (2.18), and (2.19), for our (PS) sequence {(sn, un)}, we have

O(1) ≥ (λ+ 3)Φ̃(sn, un)− ∂sΦ̃(sn, un)

≥ λ+ 3

2
s2
n − sn +

λ

2
e3sn

∫
|∇un|2 +

λ− 3

4
e6sn

(∫
|∇un|2

)2

+ (µ− λ)e3sn

∫
G(esnun) +

e5sn

2

∫
[(λ− 2)V (xesn)−∇V (xesn) · xesn ]u2

n

≥ λ+ 3

2
s2
n − sn +

λ

2
e3sn

∫
|∇un|2 +

λ− 3

4
e6sn

(∫
|∇un|2

)2

+ (µ− λ)C

∫
|vn|µ − Cλ

(∫
|vn|µ

)2/µ

,

where vn(x) = esun
(
x
es

)
. From µ > λ, 2/µ < 1 we infer (µ− λ)Cξn − Cλξn2/µ →

+∞ if ξn = |vn|2µ → +∞, we deduce from the previous estimate that

|sn|,
∫
|∇un|2,

∫
|vn|µ,

∫
G(esnun) are bounded; (2.20)

recalling (2.5) and Φ̃(sn, un) = O(1) we obtain∫
V (xesn)u2

n ≤ O(1). (2.21)



EJDE-2023/13 SCHRÖDINGER-KIRCHHOFF EQUATIONS 9

To complete the proof of the boundedness of ‖un‖, let S > 0 be such that |sn| ≤ S.
Since V ≥ −m on {V ≤ k}, we deduce∫

V (xesn)u2
n = e−2sn

∫
V (xesn)v2

n(xesn) = e−5sn

∫
V (x)v2

n

≥ e−5S
(∫
{V >k}

V (x)v2
n −m

∫
{V≤k}

v2
n

)
≥ e−5S

(∫
{V >k}

V (x)v2
n −m|{V ≤ k}|(µ−2)/µ

(∫
|vn|µ

)2/µ)
≥ e−5S

∫
{V >k}

V (x)v2
n −O(1),

where we used (A1) and (2.20) in the last inequality. From (2.21) we thus infer∫
{V >k}

V (x)v2
n ≤ O(1).

Next, k > m implies V +m ≤ 2V on {V > k} and V +m ≤ 2k on {V ≤ k}. Using
Proposition 2.1 we obtain

‖un‖2 =

∫ (
|∇un|2 + Ṽ (x)u2

n

)
=

∫
|∇un|2 + e−5sn

∫
Ṽ (xe−sn)v2

n

≤
∫
|∇un|2 + e(|κ|−5)S

∫
Ṽ (x)v2

n

≤
∫
|∇un|2 + 2e(|κ|−5)S

(∫
{V >k}

V (x)v2
n + k

∫
{V≤k}

v2
n

)
≤
∫
|∇un|2 + 2e(|κ|−5)S

(∫
{V >k}

V (x)v2
n

+ k|{V ≤ k}|(µ−2)/µ
(∫
|vn|µ

)2/µ)
≤ O(1)

by (2.20), proving the boundedness of {un} in X. Using the compact embedding
X ↪→ Ls for s ∈ [2, 6), by standard argument as in [9, Lemma 3.2], we see that {un}
has a convergent subsequence. Hence {(sn, un)} has a convergent subsequence. �

3. Critical groups and proof of Theorem 1.1

Having established the (PS) condition for Φ̃, we are now ready to present the
proof of Theorem 1.1. We start by recalling some concepts and results from infinite-
dimensional Morse theory (see e.g., Chang [5] and Mawhin and Willem [21, Chapter
8]).

Let X be a Banach space, ϕ : X → R be a C1 functional, u be an isolated critical
point of ϕ and ϕ(u) = c. Then

Ci(ϕ, u) := Hi(ϕc, ϕc \ {0}), i ∈ N = {0, 1, 2, . . .},

is called the i-th critical group of ϕ at u, where ϕc := ϕ−1(−∞, c] and H∗ stands
for the singular homology with coefficients in Z.



10 S. JIANG, L.-F. YIN EJDE-2023/13

If ϕ satisfies the (PS) condition and the critical values of ϕ are bounded from
below by α, then following Bartsch and Li [3], we define the i-th critical group of
ϕ at infinity by

Ci(ϕ,∞) := Hi(X,ϕα), i ∈ N.
By the deformation lemma, it is well known that the homology on the right-hand
side does not depend on the choice of α.

Proposition 3.1 ([3, Proposition 3.6]). If ϕ ∈ C1(X,R) satisfies the (PS) con-
dition and C`(ϕ, 0) 6= C`(ϕ,∞) for some ` ∈ N, then ϕ has a nonzero critical
point.

Proposition 3.2 ([17, Theorem 2.1]). Suppose ϕ ∈ C1(X,R) satisfies the (PS)
condition and has a local linking at 0 with respect to the decomposition X = Y ⊕Z,
i.e., for some ε > 0,

ϕ(u) ≤ 0 for u ∈ Y ∩Bε,
ϕ(u) > 0 for u ∈ (Z \ {0}) ∩Bε,

where Bε = {u ∈ X : ‖u‖ ≤ ε}. If ` = dimY <∞, then C`(ϕ, 0) 6= 0

Since Φ̃ satisfies the (PS) condition, the critical group C∗(Φ̃,∞) of Φ̃ at infinity

makes sense. To study C∗(Φ̃,∞) we need the following lemma.

Lemma 3.3. Suppose that (A1)–(A5) hold. Then there exists a constant Mλ ≥ 0
such that

d

dt
Φ̃(τ, ut) ≤

λ+ 3

t

(
Φ̃(τ, ut) +Mλ

)
(3.1)

for all t > 0, u ∈ X and τ ∈ R.

Proof. Let v = ueτ . By (2.4) and Remark 2.2 we observe that

Φ̃(τ, ut) =
τ2

2
+ Φ(uteτ ) =

τ2

2
− log2 t

2
+ Φ̃(log t, v) ≥ Φ̃(log t, v)− log2 t

2
(3.2)

d

dt
Φ̃(τ, ut) = − log t

t
+

1

t
∂sΦ̃(log t, v). (3.3)

We claim that for given λ ∈ (3, µ), there exists Mλ > 0 such that

∂sΦ̃(s, v)− s ≤ (λ+ 3)
(

Φ̃(s, v)− s2

2
+Mλ

)
(3.4)

for all s ∈ R and v ∈ X. Indeed, by (2.14), (2.18), and (2.19) we have

(λ+ 3)Φ̃(s, v)− ∂sΦ̃(s, v) ≥ λ+ 3

2
s2 − s+ (µ− λ)C

∫
|v|µ − Cλ

(∫
|v|µ

)2/µ

.

The last two terms are bounded from below because µ > λ and 2/µ < 1, thus (3.4)
holds. Then, using (3.2), (3.3), and (3.4), with s = log t and v = ueτ , we deduce
that

d

dt
Φ̃(τ, ut) = − log t

t
+

1

t
∂sΦ̃(log t, v)

≤ λ+ 3

t

(
Φ̃(log t, ueτ )− log2 t

2
+Mλ

)
=
λ+ 3

t
(Φ(uteτ +Mλ))

≤ λ+ 3

t

(
Φ̃((τ, ut) +Mλ)

)
. �



EJDE-2023/13 SCHRÖDINGER-KIRCHHOFF EQUATIONS 11

Lemma 3.4. Suppose (A2), (A4), (A5) hold. Then

lim
t→+∞

Φ̃(s, ut) = −∞

for all (s, u) ∈ R×X \ {0}.

Proof. Because ues 6= 0, and (ut)es = (ues)t, it suffices to prove that if v 6= 0 then
Φ(vt)→ −∞ as t→ +∞, see (2.4). From the proof of (2.5) we obtain

Φ(vt) =
t3

2

∫
|∇v|2 +

1

2

∫
V (x)t2v2

(x
t

)
+
t6

4

(∫
|∇v|2

)2

− t3
∫
G(tv)

Assumption (A5) implies

G(t) ≤ C|t|µ for |t| < 1, G(t) ≥ C|t|µ for |t| ≥ 1. (3.5)

Since v 6= 0, we can assume that for some ε > 0, |{|v| ≥ ε}| is finite and positive
and by (3.5) we have∫

G(tv) ≥ C
∫
|v|≥ε

tµ|v|µ ≥ Cεµ|{|v| ≥ ε}|tµ =: Cνt
µ,

for some Cν > 0 and t ≥ 1/ε.
As V is bounded on BR, it follows that∫

BR

V (x)t2v2
(x
t

)
≤ ‖V ‖L∞(BR)

∫
t2v2

(x
t

)
= t5‖V ‖L∞(BR)

∫
v2.

Assumption (A2) implies that for |ω| = R and r ≥ 1,

d

dr

( Ṽ (rω)

r

)
=
∇V (rω) · rω − Ṽ (rω)

r2
≤ 0

where Ṽ (x) = V (x) +m, so that

H(x) =
Ṽ (x)

|x|
χR3\BR(x) is radially non-increasing.

Letting w(x) = v(x)|x|1/2, we have∫
R3\BR

Ṽ t2v2
(x
t

)
= |t|3

∫
H(x)w2

(x
t

)
= |t|6

∫
H(xt)w2,

and by the monotonicity of H, H(xt)w2 ↘ 0 as t → +∞. Therefore, by the
monotone convergence theorem,∫

V (x)t2v2
(x
t

)
≤
∫
BR

V (x)t2v2
(x
t

)
+

∫
R3\BR

Ṽ t2v2
(x
t

)
≤ t5‖V ‖L∞(BR)

∫
v2 + o(t6).

In summary,

Φ(vt) ≤
t3

2

∫
|∇v|2 +

t6

4

(∫
|∇v|2

)2

+
t5‖V ‖L∞(BR)

2

∫
v2 +o(t6)−Cνtµ+3 → −∞,

as t→ +∞, because µ+ 3 > 6. �

Lemma 3.5. Suppose (A1)–(A5) hold. Then for any sufficiently negative a ∈ R,

Ci(Φ̃,∞) = 0 for all i ∈ N.



12 S. JIANG, L.-F. YIN EJDE-2023/13

Proof. Let Ẋ = X \ {0} and consider the continuous map R × Ẋ × R+ → R × Ẋ
given by

(s, u, t) 7→ (s, ut).

Choose λ ∈ (3, µ) and a < −Mλ, where Mλ ≥ 0 is given in Lemma 3.3. Then by
Lemma 3.4, we have

lim
t→+∞

Φ̃(s, ut) = −∞, ∀(s, u) ∈ R× Ẋ.

So there is ts,u > 0 such that Φ̃(s, uts,u) = a. A direct computation and Lemma
3.3 gives

d

dt

∣∣
t=ts,u

Φ̃(s, ut) ≤
λ+ 3

ts,u

(
Φ̃(s, uts,u) +Mλ

)
=
λ+ 3

ts,u
(a+Mλ) < 0.

By the implicit function theorem, given (s, u) ∈ R × Ẋ, there is a unique solution

t = T (s, u) for the equation Φ̃(s, ut) = a, and the map

T : {(s, u) ∈ R× Ẋ : Φ̃(s, u) > a} → R+

is continuous. Using the continuous function T , it is standard (see [24, 18]) to

construct a deformation from R× Ẋ to the level set Φ̃a = Φ̃−1(−∞, a], and deduce

Ci(Φ̃,∞) = Hi(R×X, Φ̃a) ∼= Hi(R×X,R× Ẋ) ∼= Hi(R×X, {0} × S∞) = 0,

for all i ∈ N, where R× Ẋ deformation retracts to {0} × S∞. �

Recalling that X+, X− and X0 are the negative, positive and null eigenspaces
of the bilinear form defined in (1.9). To study the critical group C∗(Φ̃, 0) of Φ̃ at
origin, we consider the decomposition

R×X = X̃− ⊕ X̃+

where X̃− = X− ⊕X0, X̃+ = R⊕X+.

Lemma 3.6. Assume (A1)–(A5) hold. If either

(1) dimX− > 0, dimX0 = 0,
(2) dimX0 > 0 and G(t) ≥ c|t|ν for some ν < 4,

then the functional Φ̃ has a local linking at 0 with respect to the decomposition
R×X = X̃− ⊕ X̃+.

Proof. It suffices to show that for r > 0 small enough,

Φ̃(s, u) > 0 for (s, u) ∈ X̃+, 0 < ‖(s, u)‖ < r, (3.6)

Φ̃(s, u) ≤ 0 for (s, u) ∈ X̃−, ‖(s, u)‖ < r. (3.7)

We have ∫
V (xes)u2 =

∫
V (x)u2 +

∫ (∫ 1

0

d

dτ
V (esτx)dτ

)
u2

=

∫
V (x)u2 + s

∫ (∫ 1

0

∇V (esτx) · (esτx)dτ
)
u2.



EJDE-2023/13 SCHRÖDINGER-KIRCHHOFF EQUATIONS 13

Condition (A3) and (2.2) yield

|
∫
V (esx)u2 −

∫
V (x)u2| ≤ |s|

∫ (∫ 1

0

|∇V (esτx) · (esτx)|dτ
)
u2

≤ κ|s|
∫ (∫ 1

0

Ṽ (esτx)dτ
)
u2

≤ κ|s|
∫ (∫ 1

0

eκ|s|τ Ṽ (x)dτ
)
u2

≤ (eκ|s| − 1)

∫
Ṽ (x)u2

≤ O(s)‖u‖2 = o(‖(s, u)‖2).

(3.8)

Moreover, (A4) and (A5) imply that

|G(t)| ≤ C(|t|µ + |t|p+1).

Consequently, for ‖(s, u)‖ → 0, we have

e3s
∣∣ ∫ G(esu)

∣∣ ≤ Ce3s

∫ (
|esu|µ + |esu|p+1

)
≤ o(‖u‖2), (3.9)

while by (1.7) we easily have |V (x)| ≤ 3Ṽ (x), hence

|s|
∫
|V (x)|u2 ≤ 3|s|

∫
Ṽ (x)u2 = o(‖(s, u)‖2). (3.10)

Combining (3.8)-(3.10), we obtain

Φ̃(s, u) =
s2

2
+
e3s

2

∫
|∇u|2 +

e6s

4

(∫
|∇u|2

)2

+
e5s

2

∫
V (xes)u2 − e3s

∫
G(esu)

=
s2

2
+

1 +O(s)

2

∫
|∇u|2 +

1 +O(s)

2

∫
V u2

+O(‖u‖4) + o(‖(s, u)‖2) + o(‖u‖2)

=
1

2

∫ (
|∇u|2 + V (x)u2

)
+
s2

2
+O(‖u‖4) + o(‖(s, u)‖2).

Then, it is easy to see that (3.6) is true.
Now we prove (3.7) for r > 0 small. In item (1) we have s = 0, it is easy to see

that
Φ̃(0, u) < 0 for u ∈ X−, ‖u‖ < r.

In item (2), by simple computations,

Φ̃(0, u) =
1

2

∫ (
|∇u|2 + V (x)u2

)
+

1

4

(∫
|∇u|2

)2

−
∫
G(u)

≤ 1

2

∫ (
|∇u|+ V (x)u2

)
+O(‖u‖4)− c

∫
|u|ν .

Since all norms on X− ⊕X0 are equivalent we deduce that

Φ̃(0, u) ≤ −η−‖u−‖2 +O(‖u‖4)− c‖u‖ν for u ∈ X− ⊕X0.

Because ν < 4, we can choose r > 0 small enough such that

Φ̃(0, u) < 0 for u ∈ X− ⊕X0, ‖u‖ < r.

Hence, (3.7) has also been verified item (2). �



14 S. JIANG, L.-F. YIN EJDE-2023/13

Proof of Theorem 1.1. We have shown that Φ̃ satisfies (PS) condition and has

a local linking at 0 with respect to the decomposition R × X = X̃− ⊕ X̃+. By
Proposition 3.2 we have

Ci(Φ̃, 0) 6= 0,

where i = dim X̃−. By Lemma 3.5, Ci(Φ̃,∞)=0, therefore

Ci(Φ̃, 0) 6= Ci(Φ̃,∞).

Applying Proposition 3.1, we know that Φ̃ has a nonzero critical point (s̄, ū). Fur-
thermore, Lemma 2.5 implies that s̄ = 0, ū 6= 0 and DΦ(ū) = 0. Thus Φ has a
nonzero critical point ū, which is a nontrivial solution of the problem (1.6). �

Acknowledgments. This research was partially supported by NSFC (12071387).

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Shuai Jiang

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Email address: jiangshuai0915@163.com

Li-Feng Yin
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Email address: yin136823@163.com


	1. Introduction
	2. Palais-Smale condition
	3. Critical groups and proof of Theorem 1.1
	Acknowledgments

	References