Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 18, pp. 1–11.

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

DOI: 10.58997/ejde.2024.18

EXISTENCE OF HIGH ENERGY SOLUTIONS FOR

SUPERLINEAR COUPLED KLEIN-GORDONS AND

BORN-INFELD EQUATIONS

LIXIA WANG, PINGPING ZHAO, DONG ZHANG

Communicated by Marko Squassina

Abstract. In this article, we study the system of Klein-Gordon and Born-

Infeld equations

−∆u+ V (x)u− (2ω + φ)φu = f(x, u), x ∈ R3,

∆φ+ β∆4φ = 4π(ω + φ)u2, x ∈ R3,

where ∆4φ = div(|∇φ|2∇φ), ω is a positive constant. Assuming that the

primitive of f(x, u) is of 2-superlinear growth in u at infinity, we prove the

existence of multiple solutions using the fountain theorem. Here the potential
V are allowed to be a sign-changing function.

1. Introduction and main results

In this article, we study Klein-Gordon equation using Born-Infeld theory

−∆u+ V (x)u− (2ω + φ)φu = f(x, u), x ∈ R3,

∆φ+ β∆4φ = 4π(ω + φ)u2, x ∈ R3,
(1.1)

where ω is a positive constant, V ∈ C(R3,R), and f ∈ C(R3 ×R,R). By using the
local linking theorem and the fountain theorem, we obtain multiple solutions for
(1.1).

It is well known that Klein-Gordon equation can be used in theory of electrically
charged fields [16]. The Born-Infeld theory is proposed by Born [7, 8, 9] to overcome
the infinite energy problem associated with a point-charge source in the original
Maxwell theory. The presence of the nonlinear term f simulates the interaction
between many particles or external nonlinear perturbations. For more details in
the physical aspects, we refer the readers to [5, 10, 17, 21, 30].

In recent years, the Born-Infeld nonlinear electromagnetism has become more
important since its relevance in the theory of superstring and membranes. By
using variational methods, several existence results for problem (1.1) have been
found with constant potential V (x) = m2 − ω2. Next we recall some of them.

2020 Mathematics Subject Classification. 35B33, 35J65, 35Q55.
Key words and phrases. Klein-Gordon equation; Born-Infeld theory; superlinear;

fountain theorem.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted October 30, 2023. Published February 16, 2024.

1



2 L. WANG, P. ZHAO, D. ZHANG EJDE-2024/18

In 2002, D’Avenia et al [15] considered for the Klein-Gordon equation on R3

−∆u+ [m2 − (ω + φ)2]φu = f(x, u), x ∈ R3,

∆φ+ β∆4φ = 4π(ω + φ)u2, x ∈ R3,
(1.2)

with pure power nonlinearity, i.e., f(x, u) = |u|p−2u, where ω and m are constants.
By using the mountain pass theorem, they proved that (1.2) has infinitely many
radially symmetric solutions under the assumptions that |m| > ω and 4 < p < 6.

Mugnai [21] covered the case 2 < p ≤ 4 assuming
√

p−2
2 |m| > ω > 0. Later, for

f(x, u) = |u|p−2u+ |u|2∗−2u, i.e. the critical Sobolev case was studied in [23]. The
authors obtained a nontrivial solution under the conditions 4 < p < 6 and m > ω.
The authors in [19] improved the result of [23] and studied the existence of ground
state solution. Zhang and Liu [32] considered the existence and multiplicity of
sign-changing solutions by the method of invariant sets of descending flow.

Recently, for general potential V (x), Chen and Song [14] obtained the exis-
tence of multiple nontrivial solutions for (1.1) with the nonlinearity f(x, u) =
λk(x)|u|q−2u + g(x)|u|p−2u; that is, the Klein-Gordon equation with concave and
convex nonlinearities coupled with Born-Infeld equations on R3. Other related re-
sults about homogeneous Klein-Gordon equation with Born-Infeld equations can
be found in [1, 11, 24, 25, 28, 31].

Next, we consider the non-homogeneous case, that is f(x, u) is instead of f(x, u)+
h(x). Chen and Li [12] proved that (1.1) has two nontrivial radially symmetric so-
lutions if f(x, u) = |u|p−2u and h(x) is radially symmetric. In [26], the authors
obtain the existence of two solutions by the Mountain Pass Theorem and the Eke-
land’s variational principle in critical point theory for general f(x, u). In [27], the
authors consider the existence of multiple solutions for nonhomogeneous Klein-
Gordon equation with sign-changing potential coupled with Born-Infeld theory.

Motivated by the above works, we consider system (1.1) with more general po-
tential V (x) and the primitive of f(x, u) is of 2-superlinear growth in u at infinity.
Precisely, we make the following assumptions.

(A1) V ∈ C(R3,R) is bounded below and, for every C > 0, meas{x ∈ R3 :
V (x) ≤ C} < +∞, where meas denotes the Lebesgue measures;

(A2) f ∈ C(R3 × R,R) and there exist C > 0 and p ∈ (2, 6) such that

|f(x, t)| ≤ C(1 + |t|p−1);

(A3) f(x, t) = o(t) uniformly in x as t→ 0;
(A4) f(x, t)/t→ +∞ uniformly in x as |t| → +∞;
(A5) There exists θ > 2 and b > 0 such that F(x, t) := 1

θf(x, t)t−F (x, t) ≥ −bt2,

where F (x, t) :=
∫ t

0
(x, s)ds.

The condition

(AR) There exists µ > 4 such that µF (x, t) ≤ tf(x, t), for all (x, t) ∈ R3 × R
is widely used in the studies of elliptic problem by variational methods. Condition
(AR) is used not only to prove that the Euler-Lagrange function associated has a
mountain pass geometry, but also to guarantee that the Palais-Smale sequences, or
Cerami sequences are bounded. Obviously, we can observe that the condition (AR)
implies the following condition:

(A6) There exist µ > 4 and C1, C2 > 0 such that F (x, t) ≥ C1|t|µ − C2, for t
sufficiently large.



EJDE-2024/18 COUPLED KLEIN-GORDON AND BORN-INFELD EQUATONS 3

Moreover, the condition (A6) implies condition (A4).
Another widely employed condition is the following condition, which is first in-

troduced by Jeanjean [18].

(A7) There exist θ0 ≥ 1 such that θ0F(x, t) ≥ F(x, st) for all s ∈ [0, 1] and t ∈ R,
where F(x, t) is given in (A5).

We can observe that when s = 0, then F(x, t) ≥ 0, but for our condition (A5),
F(x, t) may assume negative values. Therefore, it is interesting to consider 2-
superlinear problems under conditions (A4) and (A5).

Condition (A5) was used by Alves, Soares and Souto in [2]. With the additional
conditions that

α = inf
x∈R3

V (x) > 0 (1.3)

and b ∈ [0, α), they proved that all Cerami sequences are bounded. Under the
much weaker condition (A5), we can obtain the boundedness of Palais-Smale se-
quences, see Lemma 2.4. In 2015, Chen and Liu [13] also used conditions (A4) and
(A5) to show the existence of infinitely many solutions for Schrödinger-Maxwell
systems. In our case, many technical difficulties arise because of the presence of
the non-local term φ, which is not homogeneous as it is in the Schrödinger-Maxwell
systems. Hence, a more careful analysis of the interaction between the couple (u, φ)
is required.

By (A1), we know that V is bounded from below, hence we may choose V0 > 0
such that

Ṽ (x) := V (x) + V0 > 1, ∀x ∈ R3

and define a Hilbert space

E :=
{
u ∈ H1(R3) :

∫
R3

V (x)u2 dx <∞
}

with the inner product

〈u, v〉 =

∫
R3

(
∇u · ∇v + Ṽ (x)uv

)
dx

and the norm ‖u‖ = 〈u, u〉1/2. We also know that if V is coercive, then (A1) is
satisfied.

Obviously, the embedding E ↪→ Ls(R3) is continuous, for each s ∈ [2, 2∗]. The
norm on Ls = Ls(R3) with 1 < s <∞ is |u|s = (

∫
R3 |u|s dx)1/s. Consequently, for

each s ∈ [2, 6], there exists a constant ds > 0 such that

|u|s ≤ ds‖u‖, ∀u ∈ E. (1.4)

D(R3) is the completion of C∞0 (R3) with respect to the norm

‖u‖D := |∇u|2 + |∇u|4.

D(R3) is continuously embedded in D1,2(R3). By the Sobolev inequality, we know
that D1,2(R3) is continuously embedded in L6 = L6(R3) and D(R3) is continuously
embedded in L∞ = L∞(R3).

System (1.1) has a variational structure. In fact, we consider the functional
J : E ×D(R3)→ R defined by

J (u, φ) =
1

2

∫
R3

(|∇u|2 + V (x)u2 − (2ω + φ)φu2) dx− 1

8π

∫
R3

|∇φ|2 dx



4 L. WANG, P. ZHAO, D. ZHANG EJDE-2024/18

− β

16π

∫
R3

|∇φ|4 dx−
∫
R3

F (x, u) dx.

Solutions (u, φ) ∈ E ×D(R3) of system (1.1) are the critical points of J . As it is
pointed in [14], the functional J is strongly indefinite and is difficult to investigate.
By the reduction method described in [6], we are led to the study of a new functional
I : E → R defined by I(u) = J (u, φu). By Proposition 2.1 below, I(u) as defined
next does not present such strongly indefinite nature. Now we can state our main
result.

Theorem 1.1. Assume that (A1)—(A5) are satisfied, and f is odd in u. If 0 is not
an eigenvalue of (2.2), then (1.1) has a sequence of solutions (un, φn) ∈ E×D(R3)
such that the energy J (un, φn)→ +∞.

We emphasize that unlike all previous results about system (1.1), see e.g. [1, 11,
14, 23, 25], we do not assume that the potential is the positive constant V (x) =
m2 − ω2. We allow the potential V be sign changing. The author [20] considered
the multiplicity of solutions for Klein-Gordon-Maxwell system. There the author
assumed in addition that α = infx∈R3 V (x) > 0, and (AR) or (A6). When V is
positive, the quadratic part of the functional I (see (1.3)) is positively definite, and
I has a mountain pass geometry. Therefore, the mountain pass lemma [22] can be
applied. In our case, the quadratic part may possesses a nontrivial negative space
E−, so I no longer possesses the mountain pass geometry. Therefore the methods in
[20] cannot be applied here. To obtain our result, we adopt a technique developed
in [13].

We denote by ” ⇀ ” weak convergence, and by ”→ ” strong convergence. Also
if we take a subsequence of a sequence {un}, we shall denote it again {un}.

2. Variational setting and compactness condition

Evidently, the properties of φu play an important role in the study of J . So we
need the following technical results.

Proposition 2.1. For each u ∈ H1(R3), there exists a unique φ = φu ∈ D(R3)
which satisfies

∆φ+ β∆4φ = 4π(φ+ ω)u2 in R3.

Moreover, the map Φ : u ∈ H1(R3) 7→ φu ∈ D(R3) is continuously differentiable,
and

(i) −ω ≤ φu ≤ 0 on the set {x ∈ R3|u(x) 6= 0};
(ii)

∫
R3(|∇φu|2 + β|∇φu|4) dx ≤ 4πω2|u|22.

The first part of Proposition 2.1 was proved in [14], and the second part in [21].
After multiplying

∆φ+ β∆4φ = 4π(φ+ ω)u2

by φu and integrating by parts, by the condition (i), we obtain that∫
R3

(|∇φu|2 + β|∇φu|4) dx = −4π

∫
R3

(φu + ω)φuu
2 dx

≤ −4πω

∫
R3

φuu
2 dx

≤ 4πω2|u|22.



EJDE-2024/18 COUPLED KLEIN-GORDON AND BORN-INFELD EQUATONS 5

By Proposition 2.1 and (1.1), if u ∈ E is a critical point of I, then (u, φu) ∈
E × D(R3) is a critical point of J , that is, (u, φu) ∈ E × D(R3) is a solution of
(1.1). We can obtain a C1 functional I : E → R given by

I(u) = J (u, φu) =
1

2

∫
R3

[|∇u|2 + V (x)u2 − (2ω + φu)φuu
2] dx

− 1

8π

∫
R3

|∇φu|2 dx−
β

16π

∫
R3

|∇φu|4 dx−
∫
R3

F (x, u) dx

=
1

2

∫
R3

(|∇u|2 + V (x)u2 + φ2
uu

2) dx

+
1

8π

∫
R3

|∇φu|2 dx+
3β

16π

∫
R3

|∇φu|4 dx−
∫
R3

F (x, u) dx

=
1

2

∫
R3

(|∇u|2 + V (x)u2 − ωφuu2) dx+
β

16π

∫
R3

|∇φu|4 dx

−
∫
R3

F (x, u) dx.

(2.1)

We consider the map Φ : E → D,u→ φu. By standard arguments, Φ ∈ C1(E,D).
The Gateaux derivative of I is

〈I ′(u), v〉 =

∫
R3

(∇u · ∇v + V (x)uv − (2ω + φu)φuuv) dx−
∫
R3

f(x, u)v dx

for all u, v ∈ E.
Furthermore, under the condition (A1), the embedding E ↪→ Ls(R3) is compact

for any s ∈ [2, 6) (See [3]). By the compact embedding E ↪→ L2(R3) and the
standard elliptic theory [33], it is easy to see that the eigenvalue problem

−4u+ V (x)u = λu, u ∈ E (2.2)

possesses a complete sequence of eigenvalues

−∞ < λ1 ≤ λ2 ≤ λ3 ≤ . . . , λj → +∞.
Each λj has finite multiplicity and |λj |2 = 1. Denote ej be the eigenfunction of λj .
E− is spanned by the eigenfunctions corresponding to negative eigenvalues. Note
that the negative space E− of the quadratic part of I is nontrivial if and only if
some λj is negative.

If λ1 > 0, we can easy to prove that I has the mountain pass geometry, so we
omit this case. Since 0 is not an eigenvalue of (2.2), we assume that there exists
l ≥ 1 such that 0 ∈ (λl, λl+1). Set

E− = span{e1, . . . , el}, E+ = (E−)⊥. (2.3)

Then E− and E+ are the negative space and positive space of the quadratic form

N(u) =
1

2

∫
R3

(|∇u|2 + V (x)u2) dx

respectively, and dimE− <∞. Moreover, there is a positive constant B such that

±N(u) ≥ B‖u‖2, u ∈ E±. (2.4)

To prove Theorem 1.1, we shall use the fountain theorem by Bartsch [4]; see also
[29, Theorem 3.6]. For k = 1, 2, . . . , set

Yk = span{e1, . . . , ek}, Zk = span{ek+1, . . . , }. (2.5)



6 L. WANG, P. ZHAO, D. ZHANG EJDE-2024/18

Proposition 2.2 (Fountain theorem). Assume the even functional I ∈ C1(E,R)
satisfies the (PS) condition. If there is a positive constant K such that for any
k ≥ K there exist ρk > rk > 0 such that

(I) ak = maxu∈Yk, ‖u‖=ρk I(u) ≤ 0,
(II) bk = infu∈Zk, ‖u‖=rk I(u)→ +∞ as k → +∞,

then I has a sequence of critical points {uk} such that I(uk)→ +∞.

Proposition 2.3. Assume that p1, p2 > 1, r, q ≥ 1 and Ω ⊂ RN . Let g be a
Caratheodory function on Ω× R that satisfies

|g(x, t)| ≤ a1|t|(p1−1)/r + a2|t|(p2−1)/r, for all (x, t) ∈ Ω× R,

where a1, a2 ≥ 0. If un → u in Lp1(Ω) ∩ Lp2(Ω), and un → u a.e. x ∈ Ω, then for
each v ∈ Lp1q(Ω) ∩ Lp2q(Ω), we have

lim
n→∞

∫
Ω

|g(x, un)− g(x, u)|r|v|q dx = 0.

To study the functional I, we will write the functional I in a form in which the
quadratic part is ‖u‖2. Let h(x, t) = f(x, t)+V0t. Then, by (A5) and computations,
we obtain that

H(x, t) :=

∫ t

0

h(x, s)ds ≤ t

θ
h(x, t) + Ṽ0t

2, Ṽ0 := b+
V0

2
− V0

θ
> 0. (2.6)

By (A4) we have

lim
|t|→∞

h(x, t)t

t2
= +∞. (2.7)

Furthermore, by (A3) we obtain

lim
|t|→0

h(x, t)t

tθ
= lim
|t|→0

( t2
tθ
· f(x, t)t+ V0t

2

t2

)
= +∞.

Hence there exists M > 0 such that

h(x, t)t ≥ −Mtθ, ∀t ∈ R. (2.8)

With the modified nonlinearity h, the functional I : E → R can be rewritten in the
form

I(u) =
1

2
‖u‖2 − ω

2

∫
R3

φuu
2 dx+

β

16π

∫
R3

|∇φu|4 dx−
∫
R3

H(x, u) dx (2.9)

with derivative

〈I ′(u), v〉 = 〈u, v〉 −
∫
R3

(2ω + φu)φuuv dx−
∫
R3

h(x, u)v dx.

Lemma 2.4. Assume (A1)—(A5) are satisfied, then the function I satisfies the
(PS) condition.

Proof. It follows from 1
θ tf(x, t)− F (x, t) ≥ −bt2 that condition (A4) is equivalent

to

lim
|t|→+∞

H(x, t)

tθ
= +∞.

Let {un} be a (PS) sequence, i.e.,

I(un)→ c > 0, 〈I ′(un), un〉 → 0.



EJDE-2024/18 COUPLED KLEIN-GORDON AND BORN-INFELD EQUATONS 7

We first prove that {un} is bounded in E. Arguing by contradiction, suppose that
{un} is unbounded, passing to a subsequence, by (2.6), we obtain

θ sup
n
I(un) + ‖un‖ ≥ θI(un)− 〈I ′(un), un〉

=
(θ

2
− 1
)
‖un‖2 −

ωθ

2

∫
R3

φun
u2
n dx+

θβ

16π

∫
R3

|∇φun
|4 dx

+

∫
R3

(2ω + φun
)φun

u2
n dx+

∫
R3

(h(x, un)un − θH(x, un)) dx

≥
(θ

2
− 1
)
‖un‖2 − Ṽ0

∫
R3

u2
n dx.

(2.10)
Let vn = un/‖un‖. Then, going if necessary to a subsequence, by the compact
embedding E ↪→ L2(R3) we can assume that

vn ⇀ v0 in E,

vn → v0 in L2(R3),

vn(x)→ v0(x) a. e. in R3.

Dividing both sides of (2.10) by ‖un‖2, we have

Ṽ0

∫
R3

v2
0 dx ≥ 1 as n→∞.

Consequently, we have that v0 6= 0.
By (1.4) and (2.8), we have∫

v0=0

h(x, un)un
‖un‖θ

dx =

∫
v0=0

h(x, un)un
uθn

vθn dx

≥ −M
∫
v0=0

vθn dx

≥ −M
∫
R3

vθn dx

= −M |vn|θθ
≥ −Mdθθ > −∞.

(2.11)

For x ∈ {x ∈ R3|v0 6= 0}, we have |un(x)| → +∞ as n→∞. By (2.7) we have

h(x, un(x))un(x)

‖un‖2
=
h(x, un(x))un(x)

u2
n(x)

v2
n(x)→ +∞. (2.12)

Hence, by (2.11), (2.12) and Fatou’s lemma we obtain∫
R3

h(x, un)un
‖un‖θ

dx ≥
∫
v0 6=0

h(x, un)un
uθn

vθn(x) dx−Mdθθ → +∞. (2.13)

Hence ∫
R3

H(x, un)

‖un‖θ
dx→ +∞. (2.14)



8 L. WANG, P. ZHAO, D. ZHANG EJDE-2024/18

Since {un} is a (PS) sequence, using Proposition 2.1 and (2.13), for n large enough,
we have

cω + 1

≥ 1

‖un‖θ
(1

2
‖un‖2 −

ω

2

∫
R3

φun
u2
n dx+

3β

16π

∫
R3

|∇φun
|4 dx− I(un)

)
=

∫
R3

H(x, un)

‖un‖θ
dx→ +∞,

(2.15)

which is a contradiction. It follows that {un} is bounded in E.
Next we shall prove {un} contains a convergent subsequence. Without loss of

generality, passing to a subsequence if necessary, there exists u ∈ E such that
un ⇀ u in E. By using the embedding E ↪→ Ls(R3) are compact for any s ∈ [2, 6),
un → u in Ls(R3) for 2 ≤ s < 6 and un(x) → u(x) a.e. x ∈ R3. By (1.3) and the
Gateaux derivative of I, we can obtain that

‖un − u‖2

= 〈I ′(un)− I ′(u), un − u〉+ V0

∫
R3

(un − u)2 dx+ 2ω

∫
R3

(φunun − φuu)(un − u) dx

+

∫
R3

(h(x, un)− h(x, u))(un − u) dx+

∫
R3

(φ2
un
un − φ2

uu)(un − u) dx

By an easy computation, we obtain that

〈I ′(un)− I ′(u), un − u〉 → 0 as n→∞,∫
R3

[(φun
un − φuu)(un − u) dx+

∫
R3

[(φ2
un
un − φ2

uu)(un − u) dx→ 0

as n → +∞. Indeed, by the Hölder inequality, the Sobolev inequality and Propo-
sition 2.1, we obtain∣∣ ∫

R3

(φun − φu)(un − u)un dx
∣∣ ≤ |(φun − φu)(un − u)|2|un|2

≤ |φun
− φu|6|un − u|3|un|2

≤ C‖φun − φu‖|un − u|3|un|2,

where C is a positive constant. Since un → u in Ls(R3) for 2 ≤ s < 6, we obtain∣∣ ∫
R3

(φun
− φu)(un − u)un dx

∣∣→ 0 as n→ +∞,∣∣ ∫
R3

φu(un − u)(un − u) dx
∣∣ ≤ |φu|6|un − u|3|un − u|2 → 0 as n→ +∞.

Thus we obtain∫
R3

[(φun
un − φuu)(un − u) dx

=

∫
R3

(φun
− φu)(un − u)un dx+

∫
R3

φu(un − u)(un − u) dx→ 0

as n→ +∞. Since the sequence {φ2
un
un} is bounded in L3/2(R3), we have

|φ2
un
un|3/2 ≤ |φun

|26|un|3,



EJDE-2024/18 COUPLED KLEIN-GORDON AND BORN-INFELD EQUATONS 9

so ∣∣ ∫
R3

[(φ2
un
un − φ2

uu)(un − u) dx
∣∣ ≤ |φ2

un
un − φ2

uu|3/2|un − u|3

≤ (|φ2
un
un|3/2 + |φ2

uu|3/2)|un − u|3 → 0,

as n→ +∞.
By Proposition 2.3 and un → u in Ls(R3) for 2 ≤ s < 6, we have∫

R3

(h(x, un)− h(x, u))(un − u) dx→ 0 as n→ +∞.

Since un → u in L2(R3), we obtain that V0

∫
R3(un − u)2 dx→ 0 as n→ +∞.

Therefore ‖un − u‖ → 0 in E as n→∞. The proof is complete. �

Lemma 2.5. Let X be a finite dimensional subspace of E, then I is anti-coercive
on X, i.e.

I(u)→ −∞, as ‖u‖ → ∞, u ∈ X.

Proof. If this were not true, we can choose a sequence {un} ⊂ X and ξ is a real
number such that

‖un‖ → ∞, I(un) ≥ ξ. (2.16)

Let vn = un

‖un‖ . Since dimX <∞, going if necessary to a subsequence we have

‖vn − v0‖ → 0, vn(x)→ v0(x) a.e. in R3

for every v0 ∈ X, with ‖v0‖ = 1. Since v0 6= 0, similar to (2.14) we obtain that∫
R3

H(x, un)

‖un‖θ
dx→ +∞.

Arguing similar to (2.15), it follows from supn |I(un)| <∞ that

I(un) = ‖un‖θ
( ‖un‖2

2‖un‖θ
− ω

2‖un‖θ

∫
R3

φun
u2
n dx

+
β

16π‖un‖θ

∫
R3

|∇φun |4 dx−
∫
R3

H(x, un)

‖un‖θ
dx
)
→ −∞,

which is contradicts I(un) ≥ ξ. The proof is complete. �

Proof of Theorem 1.1. We will find a sequence of critical points {un} of I such that
I(un) → +∞. Since f(x, t) is odd in t, I is an even function. From Lemma 2.4 it
follows that I satisfies the (PS) condition. Therefore, it suffices to verify (I) and
(II) of Proposition 2.2.

Since dimYk <∞, by Lemma 2.5, we obtain the conclusion of (I).
By (A2) and (A3), we have

|f(x, t)| ≤ ε|t|+ Cε|t|p−1, |F (x, t)| ≤ ε

2
t2 +

Cε
p
|t|p,

where ε > 0 is very small. Then we have

|F (x, t)| ≤ B

2d2
2

t2 +
CB

p
|t|p, (2.17)

where B is defined in (2.4). We assume that 0 ∈ [λl, λl+1). Then if k > l, we have
that Zk ⊂ E+, where E+ is defined in (2.3). Now we have

N(u) ≥ B‖u‖2, u ∈ Zk (2.18)



10 L. WANG, P. ZHAO, D. ZHANG EJDE-2024/18

and, as in the proof of [29, Lemma3.8],

βk = sup
u∈Zk,‖u‖=1

|u|p → 0, as k →∞. (2.19)

Let rk = (Cpβpk)1/(2−p), where C is chosen as in (2.17). For u ∈ Zk ⊂ E+ with
‖u‖ = rk, φu ≤ 0, by (2.18) we deduce that

I(u) = N(u)− 1

2
ω

∫
R3

φuu
2 dx+

β

16π

∫
R3

|∇φu|4 dx−
∫
R3

F (x, u) dx

≥ B‖u‖2 − B

2d2
2

|u|22 −
CB

p
|u|pp

≥ B
(1

2
‖u‖2 −

Cβpk
p
‖u‖p

)
= B

(1

2
− 1

p2

)
(Cpβpk)2/(2−p).

Since βk → 0 and p > 2, it follows that

bk = inf
u∈Zk,‖u‖=rk

I(u)→ +∞.

We obtain the conclusion of (II). The proof is complete. �

Acknowledgments. This research was partially supported by the Scientific Re-
search Program of Tianjin Education Commission (2023KJ216).

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Lixia Wang

School of Sciences, Tianjin Chengjian University, Tianjin 300384, China
Email address: wanglixia0311@126.com

Pingping Zhao

School of Sciences, Tianjin Chengjian University, Tianjin 300384, China
Email address: ppzhao math@126.com



12 L. WANG, P. ZHAO, D. ZHANG EJDE-2024/18

Dong Zhang (corresponding author)

School of Sciences, Tianjin Chengjian University, Tianjin 300384, China

Email address: zhangdongtg@126.com


	1. Introduction and main results
	2. Variational setting and compactness condition
	Acknowledgments

	References