Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 74, pp. 1–11.

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

TWO SOLUTIONS FOR NONHOMOGENEOUS KLEIN-GORDON

EQUATIONS COUPLED WITH BORN-INFELD TYPE

EQUATIONS

LIXIA WANG, CHUNLIAN XIONG, PINGPING ZHAO

Abstract. This article concerns the nonhomogeneous Klein-Gordon equation

coupled with a Born-Infeld type equation,

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

∆φ+ β∆4φ = 4π(ω + φ)u2, x ∈ R3,

where ω is a positive constant. We obtain the existence of two solutions using
the Mountain Pass Theorem, and the Ekeland’s variational principle in critical

point theory.

1. Introduction and main results

We consider a nonhomogeneous Klein-Gordon equation coupled with a Born-
Infeld type equation,

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

∆φ+ β∆4φ = 4π(ω + φ)u2, x ∈ R3,
(1.1)

where ω is a positive constant. Under certain assumptions on V , f and h, we
show the existence of two solutions by using the Mountain Pass Theorem and the
Ekeland’s variational principle in critical point theory.

It is well known that the Klein-Gordon equation (1.1) can be used in the theory
of electrically charged fields (see [16]), and that the Born-Infeld theory was proposed
by Born [6, 7, 8] to overcome the infinite energy problem associated with a point-
charge source in the original Maxwell theory. The presence of the nonlinear term f
in (1.1) can model the interaction between many particles, or the external nonlinear
perturbations. For more details and physical aspects of (1.1), we refer the readers
to [4, 9, 17, 21, 28].

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. We recall some of them.

The case of h ≡ 0, that is the homogeneous case, has been widely studied. In
2002, D’Avenia et al. [14] considered for the following Klein-Gordon equation with

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

Mountain pass theorem; Ekeland’s variational principle.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted August 2, 2022. Published November 15, 2022.

1



2 L. WANG, C. XIONG, P. ZHAO EJDE-2022/74

an Born-Infeld 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)

for the power nonlinearity 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 |m| > |ω| and 4 < p < 6. Mugnai [21] studied the

case 2 < p ≤ 4 assuming
(
(p − 2)/2

)1/2|m| > ω > 0. Later, Teng et al. [22]

obtained a nontrivial solution for (1.2) with f(x, u) = |u|p−2u+ |u|2∗−2u under the
conditions 4 ≤ p < 6 and m > ω. He et al. [20] improved the existence results from
[22] and studied the existence of ground state solution for the problem(1.2) with
f(x, u) = |u|p−2u+ |u|2∗−2u.

Recently, for general potential V (x), Chen and Song [13] obtained the exis-
tence of multiple nontrivial solutions for (1.2) 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, 23, 24, 25, 29]. For the nonhomogeneous case, h 6≡ 0, Chen
and Li [10] proved that (1.1) has two nontrivial radially symmetric solutions if
f(x, u) = |u|p−2u and h(x) is radially symmetric.

Motivated by above works, we consider (1.1) with general assumptions on f ,
and without any radially symmetric assumptions on f and h. More precisely, we
assume the following:

(H1) V ∈ C(R3,R) satisfies V0 = infx∈R3 V (x) > 0. For every M > 0, and
meas{x ∈ R3 : V (x) ≤ M} < +∞, where meas denotes the Lebesgue
measures in R3;

(H2) f ∈ C(R3 × R,R) and there exist C1 > 0 and p ∈ (2, 6) such that

|f(x, t)| ≤ C1(|t|+ |t|p−1);

(H3) f(x, t) = o(t) uniformly in x as |t| → 0, uniformly for x ∈ R3;
(H4) There exist θ > 2 and D1, D2 > 0 such that F (x, t) ≥ D1|t|θ −D2, for a.e.

x ∈ R3 and every t sufficiently large;
(H5) There exist C2, r0 are two positive constants and µ > 2 such that

F(x, t) :=
1

µ
f(x, t)t− F (x, t) ≥ −C2|t|2, |t| ≥ r0;

(H6) h ∈ L2(R3), h(x) ≥ 0, h(x) 6≡ 0.

Before stating our main results, we introduce some notation. Let H1(R3) be the
usual Sobolev space endowed with the standard scalar product and norm

(u, v)H =

∫
R3

(∇u∇v + uv) dx, ‖u‖H =
(∫

R3

(|∇u|2 + |u|2) dx
)1/2

.

The space D1,2(R3) is the completion of C∞0 (R3) with respect to the norm

‖u‖D1,2(R3) =
(∫

R3

|∇u|2 dx
)1/2

.



EJDE-2022/74 KLEIN-GORDON-MAXWELL SYSTEM 3

The norm on Ls = Ls(R3) with 1 < s <∞ is given by |u|ss =
∫
R3 |u|s dx. 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 D(R3) is continuously embedded in L∞ = L∞(R3).

Under condition (H1), we define a new Hilbert space

E :=
{
u ∈ H1(R3) :

∫
R3

(|∇u|2 + V (x)u2) dx <∞
}

with the inner product

〈u, v〉 =

∫
R3

(∇u · ∇v + V (x)uv) dx

and the norm ‖u‖ = 〈u, u〉1/2. Obviously, the embedding E ↪→ Ls(R3) is continu-
ous, for 2 ≤ s ≤ 6. Consequently, for 2 ≤ s ≤ 6, there exists a constant ds > 0 such
that

|u|s ≤ ds‖u‖, ∀u ∈ E. (1.3)

Furthermore, it follows from (H1) that the embedding E ↪→ Ls(R3) is compact for
any s ∈ [2, 6) (See [3]).

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

− β

16π

∫
R3

|∇φ|4 dx−
∫
R3

F (x, u) dx−
∫
R3

h(x)u dx.

The solutions (u, φ) ∈ E × D(R3) of system (1.1) are the critical points of J .
As it is pointed in [13], the functional J is strongly indefinite and is difficult to
investigate. By the reduction method described in [5], 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) is defined as follows which does not present such strongly indefinite nature.
We can obtain a C1 functional I : E → R given by

I(u) =
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−
∫
R3

h(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−
∫
R3

h(x)u dx

=
1

2

∫
R3

(|∇u|2 + V (x)u2 − ωφuu2) dx

+
β

16π

∫
R3

|∇φu|4 dx−
∫
R3

F (x, u) dx−
∫
R3

h(x)u dx.

(1.4)



4 L. WANG, C. XIONG, P. ZHAO EJDE-2022/74

We consider the map Φ : E → D, u 7→ φu. Then 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−
∫
R3

h(x)v dx

for all u, v ∈ E. Our main result reads as follows.

Theorem 1.1. Suppose (H1)–(H6) hold. Then there exists a positive constant m0

such that system (1.1) admits at least two different solutions u0, ũ0 in E satisfying
I(u0) < 0 and I(ũ0) > 0 if |h|2 < m0.

Remark 1.2. Chen and Song [11] proved that Klein-Gordon-Maxwell equation
had two nontrivial solutions if f(x, t) satisfies the local (AR) condition:

(LAR) There exist µ > 2 and r0 > 0 such that F(x, t) := 1
µf(x, t)t − F (x, t) ≥ 0

for every x ∈ R3 and |t| ≥ r0, where F (x, t) =
∫ t

0
f(x, s)ds.

This condition is employed 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. Under condition (LAR) in this
article, F(x, t) may have negative values.

Another widely used condition was introduced by Jeanjean [18],

• There exists θ ≥ 1 such that θF1(x, t) ≥ F1(x, st) for all s ∈ [0, 1] and
t ∈ R, where F1(x, t) := 1

4f(x, t)t− F (x, t).

We can observe that if s = 0, then F1(x, t) ≥ 0, but in our condition (H5), F(x, t)
may assume negative values.

In [2, 12], the authors studied the Schrödinger-Poisson equation by assuming the
following global condition to replace the (AR) condition:

• There exists 0 ≤ β < α such that tf(t)−4F (t) ≥ −βt2, for all t ∈ R, where
α is a positive constant such that α ≤ V (x).

Notice that we only need the local condition (H5) to obtain nontrivial solutions.
Li and Tang [19] used the following condition to obtain infinitely many solutions

for homogenous KGM systems:

• There exist two positive constants D3 and r0 such that 1
4f(x, t)t−F (x, t) ≥

−D3|t|2, if |t| ≥ r0.

Obviously, our condition (H5) is weaker than this condition. Therefore, it is inter-
esting to consider the nonhomogeneous system (1.1) under the conditions (H4) and
(H5).

To the best of our knowledge, Theorem 1.1 is the first result about the existence
of two solutions for the nonhomogeneous Klein-Gordon equation coupled with Born-
Infeld equations on R3 with general nonlinearity f .

Throughout this article, letters Ci, di, Li,Mi, i = 1, 2, 3 . . . will denote various
positive constants which may vary from line to line and are not essential to the
problem. We denote the weak convergence by “⇀” and the strong convergence by
“→”. Also if we take a subsequence of a sequence {un}, we shall denote it again
by {un}.



EJDE-2022/74 KLEIN-GORDON-MAXWELL SYSTEM 5

The paper is organized as follows. In Section 2, we introduce the variational
setting for the problem and give some related preliminaries. In Section 3, we prove
our main result.

2. Variational setting and compactness condition

From [5], we know that the signs of ω is not relevant for the existence of solutions,
so we can assume that ω > 0. Evidently, the properties of φu plays an important
role in the study of J . So we need the following technical results.

Proposition 2.1. For any 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

−ω ≤ φu ≤ 0 on {x ∈ R3|u(x) 6= 0}; (2.1)∫
R3

(|∇φu|2 + β|∇φu|4) dx ≤ 4πω2|u|22. (2.2)

Inequality (2.1) was proved in [13], and (2.1) was proved in [21]. 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 system (1.1). Now we prove
the function I has the mountain pass geometry.

Lemma 2.2. Let h ∈ L2(R3) and assume that (H1)–(H3) hold. Then there are
positive constants ρ, α,m0 such that I(u) ≥ α for all u ∈ E satisfying ‖u‖ = ρ and
h satisfying |h|2 < m0.

Proof. By (H3), for any ε > 0, there exists δ > 0 such that |f(x, t)| ≤ ε|t| for all
x ∈ R3 and |t| ≤ δ. By (H2), we obtain

|f(x, t)| ≤ C1(|t|+ |t|p−1)

≤ C1

(
|t|| t

δ
|p−2 + |t|p−1

)
= C1

( 1

δp−2
+ 1
)
|t|p−1, for |t| ≥ δ, a.e. x ∈ R3.

Then for all t ∈ R and a.e. x ∈ R3 we have

|f(x, t)| ≤ ε|t|+ C1

( 1

δp−2
+ 1
)
|t|p−1 := ε|t|+ Cε|t|p−1

and

|F (x, t)| ≤ ε

2
|t|2 +

Cε
p
|t|p. (2.3)

Therefore, by (2.3), Proposition 2.1 and Hölder’s inequality, we have

I(u) ≥ 1

2
‖u‖2 − ε

2

∫
R3

|u|2 dx− Cε
p

∫
R3

|u|p dx− |h|2|u|2

≥ 1

2
‖u‖2 − ε

2
d2

2‖u‖2 −
Cε
p
dpp‖u‖p − d2|h|2‖u‖

= ‖u‖
{(1

2
− ε

2
d2

2

)
‖u‖ − Cε

p
dpp‖u‖p−1 − d2|h|2

}
.



6 L. WANG, C. XIONG, P. ZHAO EJDE-2022/74

Let ε = 1/2d2
2 and g(t) = t

4 −
Cε

p d
p
pt
p−1 for t ≥ 0. Because 2 < p < 6, there exists a

positive constant ρ such that m̃0 := g(ρ) = maxt≥0 g(t) > 0. Taking m0 := m̃0/2d
2
2,

it follows that there exists a positive constant α such that I(u)|‖u‖=ρ ≥ α for all h
satisfying |h|2 < m0. The proof is complete. �

Lemma 2.3. Assume that (H1)–(H5) are satisfied. Then there exists a function
u0 ∈ E with ‖u0‖ > ρ such that I(u0) < 0, where ρ is given in Lemma 2.2.

Proof. By (H4), there exist L1 > 0 large enough and M1 > 0, such that

F (x, t) ≥M1|t|θ, for |t| ≥ L1. (2.4)

By (2.3), we obtain

|F (x, t)| ≤ C3(1 + |t|p−2)|t|2, where C3 = max{ε
2
,
Cε
p
}, (2.5)

and then

|F (x, t)| ≤ C3(1 + Lp−2
1 )|t|2, when |t| ≤ L1. (2.6)

By (2.4) and (2.6), we have

F (x, t) ≥M1|t|θ −M2|t|2, ∀t ∈ R, (2.7)

where M2 = M1L
θ−2
1 + C3(1 + Lp−2

1 ).
Thus, by Proposition 2.1, taking u ∈ E, u 6= 0 and θ > 2 we have

I(tu) =
t2

2
‖u‖2 − t2

2

∫
R3

(2ωφtuu
2 + φ2

tuu
2) dx− 1

8π

∫
R3

|∇φtu|2 dx

− β

16π

∫
R3

|∇φtu|4 dx−
∫
R3

F (x, tu) dx− t
∫
R3

h(x)u dx

≤ t2

2
‖u‖2 +

t2

2

∫
R3

ω2u2 dx−M1t
θ

∫
R3

|u|θ dx

+M2t
2

∫
R3

u2 dx− t
∫
R3

h(x)u dx,

thus I(tu)→ −∞ as t→ +∞. The lemma is proved by taking u0 = t0u with t0 > 0
large enough and u 6= 0. �

Lemma 2.4. Under assumptions (H1)–(H6), every sequence {un} ⊂ E satisfying

I(un)→ c > 0, 〈I ′(un), un〉 → 0

is bounded in E. Moreover, {un} has a strongly convergent subsequence in E.

Proof. To prove the boundedness of {un}, we argue by contradiction. Suppose that,
up to subsequences, we have ‖un‖ → +∞ as n → +∞. Let vn = un

‖un‖ , then {vn}
is bounded. Going if necessary to a subsequence, for some v ∈ E, we obtain that

vn ⇀ v in E,

vn → v in Ls(R3), 2 ≤ s < 6,

vn(x)→ v(x) a.e. in R3.



EJDE-2022/74 KLEIN-GORDON-MAXWELL SYSTEM 7

Let Λ = {x ∈ R3 : v(x) 6= 0}. Suppose that meas(Λ) > 0, then |un(x)| → +∞ as
n→∞ for a.e. x ∈ Λ. By (1.3) and (2.7), we obtain∫

R3

F (x, un)

‖un‖θ
dx ≥M1

∫
R3

|vn|θ dx−M2
|un|2

‖un‖θ

≥M1

∫
R3

|vn|θ dx−
M2d

2
2

‖un‖θ−2

→M1

∫
Λ

|v|θ dx > 0 as n→∞.

(2.8)

By Proposition 2.1, from (1.3) and θ > 2, so we obtain that∣∣ ∫
R3

ωφunu
2
n

‖un‖θ
dx
∣∣ ≤ ω2|un|22

‖un‖θ
≤ ω2d2

2

‖un‖θ−2
→ 0 as n→∞.

∣∣ ∫
R3

β|∇φun |4

‖un‖θ
dx
∣∣ ≤ 4πω2|un|22

‖un‖θ
≤ 4πω2d2

2

‖un‖θ−2
→ 0 as n→∞.

Since h ∈ L2(R3), we obtain that∣∣ ∫
R3

h(x)un
‖un‖θ

dx
∣∣ ≤ |h|2|un|2

‖un‖θ
≤ |h|2d2

‖un‖θ−1
→ 0 as n→∞.

By the definition of I, (2.7), and the above inequalities, we have

0 = lim
n→+∞

I(un)

‖un‖θ

= lim
n→+∞

[ 1

2‖un‖θ−2
−
∫
R3

ωφun
u2
n

2‖un‖θ
dx+

β

16π

∫
R3

|∇φun
|4

‖un‖θ
dx

−
∫
R3

F (x, un)

‖un‖θ
dx−

∫
R3

h(x)un
‖un‖θ

dx
]
< 0,

which is a contradiction. Therefore, meas(Λ) = 0, which implies v(x) = 0 for
almost everywhere x ∈ R3. By (H2) and (2.5), we have that for all x ∈ R3 and
|t| ≤ r0,

|f(x, t)t− µF (x, t)| ≤ |f(x, t)t|+ µ|F (x, t)|
≤ C1(|t|2 + |t|p) + µC3(1 + |t|p−2)t2

≤ C6(1 + |t|p−2)t2

≤ C6(1 + rp−2
0 )t2,

where C6 := 2 max{C1, µC3}. From this and (H5), we obtain

f(x, t)t− µF (x, t) ≥ −C7t
2, ∀(x, t) ∈ R3 × R. (2.9)

Since h ∈ L2(R3), we can also obtain∣∣ ∫
R3

h(x)un
‖un‖2

dx
∣∣ ≤ |h|2|un|2

‖un‖2
≤ |h|2d2

‖un‖
→ 0 as n→∞. (2.10)

By (2.9), (2.10), Proposition 2.1, v = 0, and µ > 2, we have

µI(un)− 〈I ′(un), un〉
‖un‖2

=
(µ

2
− 1
)

+

∫
R3

f(x, un)un − µF (x, un)

‖un‖2
dx+

(µ
2

+ 1
) ∫

R3

φunu
2
n

‖un‖2
dx



8 L. WANG, C. XIONG, P. ZHAO EJDE-2022/74

+ 2ω

∫
R3

φun
u2
n

‖un‖2
dx+

µ

8π

∫
R3

|∇φun
|2

‖un‖2
dx+

3βµ

16π

∫
R3

|∇φun
|4

‖un‖2
dx

+ (1− µ)

∫
R3

h(x)un
‖un‖2

dx

≥
(µ

2
− 1
)
− C7|vn|22 − 2ω2|vn|22 + (1− µ)

∫
R3

h(x)un
‖un‖2

dx

≥
(µ

2
− 1
)
− (C7 + 2ω2)|vn|22 + (1− µ)

∫
R3

h(x)un
‖un‖2

dx

→ µ

2
− 1 as n→∞.

Then we have 0 ≥ 1
2 −

1
µ , which contradicts with µ > 2. Therefore {un} is a

bounded in E.
Now 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 (1.4) and the
Gateaux derivative of I, we obtain∫

R3

(|∇(un − u)|2 + V (x)(un − u)2) dx

= 〈I ′(un)− I ′(u), un − u〉+ 2ω

∫
R3

(φun
un − φuu)(un − u) dx

+

∫
R3

(f(x, un)− f(x, u))(un − u) dx+

∫
R3

(φ2
un
un − φ2

uu)(un − u) dx.

Doing easy computations we have

〈I ′(un)− I ′(u), un − u〉 → 0 as n→∞

and ∫
R3

[(2ω + φun
)φun

un − (2ω + φu)φuu](un − u) dx

= 2ω

∫
R3

[(φunun − 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 have∣∣ ∫

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‖D|un − u|3|un|2,

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



EJDE-2022/74 KLEIN-GORDON-MAXWELL SYSTEM 9

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→ +∞. In view of that the sequence {φ2
un
un} is bounded in L3/2(R3), since

|φ2
un
un|3/2 ≤ |φun

|26|un|3,

it follows that∣∣ ∫
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→ +∞.
From the convergence un → u in Ls(R3) for 2 ≤ s < 6, we have∫

R3

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

Therefore,

‖un − u‖2

= 〈I ′(un)− I ′(u), un − u〉+ 2ω

∫
R3

(φunun − φuu)(un − u) dx

+

∫
R3

(f(x, un)− f(x, u))(un − u) dx+

∫
R3

(φ2
un
un − φ2

uu)(un − u) dx→ 0

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

3. Proof of main result

Proof of Theorem 1.1. Firstly, we prove that there exists a function u0 ∈ E such
that I ′(u0) = 0 and I(u0) < 0. Since h ∈ L2(R3), h ≥ 0 and h 6≡ 0, we can choose
a function ϕ ∈ E such that ∫

R3

h(x)ϕ(x) dx > 0.

Hence, by Proposition 2.1, θ > 2 and (2.7), we obtain that

I(tϕ) ≤ t2

2
‖ϕ‖2 + t2

∫
R3

ω2ϕ2 dx−M1t
θ|ϕ|θθ +M2t

2|ϕ|22 − t
∫
R3

h(x)ϕdx < 0

for t > 0 small enough. Thus, we obtain

c0 = inf{I(u) : u ∈ Bρ} < 0,

where ρ > 0 is given by Lemma 2.2, Bρ = {u ∈ E : ‖u‖ < ρ}. By the Ekeland’s

variational principle [15], there exists a sequence {un} ⊂ Bρ such that

c0 ≤ I(un) < c0 +
1

n
,

and

I(v) ≥ I(un)− 1

n
‖v − un‖



10 L. WANG, C. XIONG, P. ZHAO EJDE-2022/74

for all v ∈ Bρ. Then by a standard procedure, we can prove that {un} is a bounded
(PS) sequence of I. Hence, by Lemma 2.4 we know that there exists a function
u0 ∈ E such that I ′(u0) = 0 and I(u0) = c0 < 0.

Secondly, we prove that there exists a function ũ0 ∈ E such that I ′(ũ0) = 0 and
I(ũ0) > 0.

By Lemmas 2.2 and 2.3 and the Mountain Pass Theorem [26, 27], there is a
sequence {un} ⊂ E such that

I(un)→ c̃0 > 0, and I ′(un)→ 0.

In view of Lemma 2.4, we know that {un} has a strongly convergent subsequence
(still denoted by {un}) in E. So there exists a function ũ0 ∈ E such that {un} → ũ0

as n→∞ and I ′(ũ0) = 0 and I(ũ0) > 0. The proof is complete. �

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

Email address: wanglixia0311@126.com

Chunlian Xiong (corresponding author)

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

Email address: xiongcltg@126.com

Pingping Zhao

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


	1. Introduction and main results
	2. Variational setting and compactness condition
	3. Proof of main result
	References