Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 01, pp. 1–14.

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

MULTIPLE POSITIVE SOLUTIONS FOR NONHOMOGENEOUS

SCHRÖDINGER-POISSON SYSTEMS WITH

BERESTYCKI-LIONS TYPE CONDITIONS

LAN-XIN HUANG, XING-PING WU, CHUN-LEI TANG

Communicated by Claudianor O. Alves

Abstract. In this article, we consider the multiplicity of solutions for non-

homogeneous Schrödinger-Poisson systems under the Berestycki-Lions type

conditions. With the aid of Ekeland’s variational principle, the mountain pass
theorem and a Pohožaev type identity, we prove that the system has at least

two positive solutions.

1. Introduction and statement of main results

In this article, we study the Schrödinger-Poisson system

−∆u+ λφu = g(u) + h(x) in R3,

−∆φ = u2 in R3,
(1.1)

where λ > 0 is a parameter, g ∈ C(R,R) and h ∈ L2(R3). System (1.1) is also
called Schrödinger-Maxwell equations, and arises in an interesting physical context.
In fact, according to a classical model, the interaction of a charged particle with
an electromagnetic field can be described by coupling the nonlinear Schrödinger’s
and Poisson’s equations (we refer the reader to previous studies [4, 5, 21] and the
references therein for more details on the physical aspects).

If h(x) ≡ 0, system (1.1) becomes the classical Schrödinger-Poisson system

−∆u+ λφu = g(u) in R3,

−∆φ = u2 in R3,
(1.2)

which was introduced by Benci and Fortunato [4]. System (1.2) has been extensively
studied under various hypotheses on the nonlinearity, see for example [1, 2, 3, 8,
9, 13, 19, 20, 22, 26, 27]. The case g(u) = |u|p−2u− u, p ∈ (2, 6) has been studied
in [3, 8, 9, 20]. D’Aprile and Mugnai [8] established the existence of a nontrivial
radial solution for p ∈ [4, 6). On the other hand, the non-radial solution of system
(1.2) was considered in [9] for p ∈ (4, 6). Ruiz [20] showed that system (1.2)
has no solution when p ∈ (2, 3] and obtained a positive radial solution by using
a constrained minimization method when p ∈ (3, 6). Azzollini and Pomponio [3]

2010 Mathematics Subject Classification. 35A15, 35B09, 35B50, 35D30.
Key words and phrases. Nonhomogeneous Schrödinger-Poisson system; variational methods;

multiple positive solutions; Berestycki-Lions type conditions.
c©2021 Texas State University.

Submitted April 1, 2020. Published January 7, 2021.
1



2 L.-X. HUANG, X.-P. WU, C.-L. TANG EJDE-2021/01

considered the ground state solutions of system (1.2) for p ∈ (3, 6). For the case
that g is a general nonlinear term, we refer the reader to [1, 2, 22, 26]. Sun and Ma
[22] dealt with the existence of a ground state solution for system (1.2) involving
a 3-superlinear nonlinearity g(u) satisfying the (AR) type condition, namely, there
exists µ > 3 such that g(u)u ≥ µG(u) > 0 for all u ∈ R\{0}. After that, the
authors in [26] improved the result of [22], who discussed the case that g(u) is
asymptotically 2-linear and obtained a ground state solution. Azzollini et al [2]
assumed that g ∈ C(R,R) and g satisfies the following conditions:

(A1) −∞ < lim inft→0 g(t)/t ≤ lim supt→0 g(t)/t = −m < 0,
(A2) −∞ < lim sup|t|→∞ g(t)/|t|4t ≤ 0,

(A3) there exists ζ > 0 such that G(ζ) :=
∫ ζ

0
g(s)ds > 0.

They showed that system (1.2) admits a nontrivial radial solution for λ small
enough. To guarantee the boundedness of Palais–Smale sequence in [2], the au-
thors used a truncation argument in [16] and Struwe’s monotonicity trick. By the
way, (A1)–(A3) are known as the Berestycki-Lions conditions, introduced in [6].
There, the authors showed that (A1)–(A3) are “almost” necessary for the existence
of nontrivial solutions to system (1.2) when λ = 0. In the sequel, Azzollini [1]
assumed that g satisfies (A1)–(A3) and obtained a nontrivial non-radial solution
for system (1.2) by using a concentration and compactness argument.

Next, we consider the nonhomogeneous case of system (1.1), that is h(x) 6≡
0. Salvatore [21] proved the existence of three radially symmetric solutions to
system (1.1) with g(u) = |u|p−2u − u, p ∈ (4, 6). Subsequently, Jiang et al [17]
discussed system (1.1) for p ∈ (2, 6) and obtained two radial solutions when |h|2 is
small enough. Particularly, Zhang et al [28] studied the following nonhomogeneous
Schrödinger-Poisson system

−∆u+Ku+ λφf(u) = g(u) + h(x) in R3,

−∆φ = 2λF (u) in R3,
(1.3)

where λ ≥ 0, K is a positive constant. They assumed the following conditions:

(A4) f ∈ C(R,R+), there exist C > 0, α ∈ (2, 4) such that f(t) ≤ C(|t| + |t|α),
t ∈ R,

(A5) g ∈ C(R,R+), there exist C > 0, p ∈ (2, 6) such that g(t) ≤ C(|t|+ |t|p−1),
t ∈ R,

(A6) limt→0+
g(t)
t = 0,

(A7) limt→∞
g(t)
t = l, where K < l ≤ ∞,

(A8) (x · ∇h) ∈ L2(R3) is nonnegative, h ∈ C1(R3) ∩ L2(R3) is a nonnegative
radial function and there exists M > 0 such that |h|2 ≤M .

Theorem 1.1 ([28]). If (A4)–(A8) are satisfied, then there exists λ0 > 0 such that
system (1.3) has at least two positive radial solutions for λ ∈ [0, λ0).

Since the condition (A7) implies that g is asymptotically linear or superlinear
at infinity and g does not satisfy the (AR) condition, it is not easy to obtain a
bounded Palais-Smale sequence. To overcome this difficulty, the authors in [28] used
the method based on Struwe’s monotonicity trick and cut-off function. For more
results on the nonhomogeneous case, see [7, 10, 11, 18, 23, 24] and the references
therein. However, in these papers, the hypotheses of nonlinear term are much
stronger than the Berestycki-Lions type conditions.



EJDE-2021/01 NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS 3

Inspired by the above works, especially by the results in [2, 17, 21, 28], we
try to give the weakest conditions on g. Then, the purpose of this article is to
obtain multiple positive solutions of system (1.1) under the Berestycki-Lions type
conditions when h(x) 6≡ 0 and λ is small enough. To state our results, we assume
that h ∈ L2(R3) is a radial function, h(x) 6≡ 0, satisfies (A1) and (A3), and the
following two conditions are satisfied.

(A9) lim|t|→∞
g(t)
|t|4t = 0,

(A10) (x · ∇h) ∈ L 6
5 (R3), where the gradient ∇h is in the weak sense.

Now we state our main results.

Theorem 1.2. Suppose that h ∈ L2(R3) is a radial function and h(x) 6≡ 0. Let
(A1), (A3), (A9), (A10) hold. Then there exist λ0,Λ > 0 such that (1.1) admits
two nontrivial radial solutions for λ ∈ (0, λ0), |h|2 < Λ.

Moreover, if h satisfies an additional condition, we can prove the solution is
positive.

Corollary 1.3. If h(x) ≥ 0 in R3 and the assumptions of Theorem 1.2 hold, then

there exist λ̃0, Λ̃ > 0 such that system (1.1) admits two positive radial solutions for

λ ∈ (0, λ̃0), |h|2 < Λ̃.

Remark 1.4. (1) From the condition (A10), our h can change its sign and (x ·∇h)
belongs to a class of functions which are different from the ones that appear in
(A8).

(2) Notice that, (A3) is much weaker than (A7) of Theorem 1.1. In fact, there

exist many functions that satisfy (A3) but not (A7), for example, g(t) = −t+ 4t3

1+t4 .

It is not difficult to verify g satisfying (A1), (A3) and (A9), however, setting K = 1
in system (1.3) we have

lim
t→∞

g(t) + t

t
= lim
t→∞

4t2

1 + t4
= 0 < K,

which implies that g does not satisfy (A7). Hence, Corollary 1.3 improves Theorem
1.1 and thus generalizes [17, Theorems 1.1, 1.2] and [21, Theorem 1.2].

It also should be pointed out that our methods are different from ones in [28],
since our g does not satisfy g ∈ C(R,R+).

This article is organized as follows. In section 2, with the aid of Ekeland’s vari-
ational principle that the first radial solution is a local minimizer u1 with negative
energy, Theorem 2.3. In section 3, we find the second radial solution u2 with pos-
itive energy by Theorem 3.6 to complete the proof of Theorem 1.2, and finish the
proof of Corollary 1.3.

Throughout this paper, we use the following notation:

• Lp(R3) denotes the Lebesgue space with the usual norm |u|p =
( ∫

R3 |u|p dx
)1/p

.

• H1(R3) is the Hilbert space endowed with the norm ‖u‖ =
( ∫

R3 |∇u|2 +u2 dx
)1/2

.

• D1,2(R3) is the completion of C∞0 (R3) with the norm ‖u‖D =
(∫

R3 |∇u|2 dx
) 1

2 .

• The best Sobolev constant of the embedding D1,2(R3) ↪→ L6(R3) is defined by

S = inf
u∈D1,2(R3)\{0}

∫
R3 |∇u|2 dx

(
∫
R3 |u|6 dx)1/3

.



4 L.-X. HUANG, X.-P. WU, C.-L. TANG EJDE-2021/01

• For each p ∈ (2, 6), there exists Sp such that |u|p ≤ Sp‖u‖ for all u ∈ H1(R3).
• u+ = max{u, 0}, u− = min{u, 0}; (H∗, ‖ · ‖∗) denotes the dual space of (H, ‖ · ‖).
• Sr := {u ∈ H1(R3) : ‖u‖ = r} and Br := {u ∈ H1(R3) : ‖u‖ < r}.
• C, Ci, ai are various positive constants.
• on(1) is a quantity tending to 0 as n→∞.

2. A weak solution with negative energy

We recall that, for every u ∈ H1(R3), the Lax-Milgram theorem implies that
there exists a unique φu ∈ D1,2(R3) such that

−∆φu = u2. (2.1)

Furthermore, we can write the integral expression for φu:

φu(x) =
1

4π

∫
R3

u2(y)

|x− y|
dy.

For more properties of φu, we refer the reader to [2]. It follows from (2.1), the
Hölder and Sobolev inequalities that

‖φu‖2D = −
∫
R3

φu∆φu dx =

∫
R3

φuu
2 dx ≤ |φu|6|u|212/5 ≤ S

−1/2‖φu‖D|u|212/5,

then there exists a constant a1 = S−1S4
12/5 > 0 such that∫

R3

φuu
2 dx ≤ S−1/2‖φu‖D|u|212/5 ≤ S

−1|u|412/5 ≤ S
−1S4

12/5‖u‖
4 = a1‖u‖4. (2.2)

In this article, we consider system (1.1) in H1
r (R3). Define the functional Iλ :

H1
r (R3)→ R by

Iλ(u) =
1

2

∫
R3

|∇u|2 dx+
λ

4

∫
R3

φuu
2 dx−

∫
R3

G(u) dx−
∫
R3

h(x)u dx.

It is standard to prove that Iλ is of class C1 whose derivative is given by

〈I ′λ(u), v〉 =

∫
R3

∇u · ∇v dx+ λ

∫
R3

φuuv dx−
∫
R3

g(u)v dx−
∫
R3

h(x)v dx,

for all v ∈ H1
r (R3). Then, if u is a critical point of Iλ, the couple (u, φu) is a

solution of (1.1). For simplicity, in many cases we just say that u ∈ H1
r (R3),

instead of (u, φu) ∈ H1
r (R3)×D1,2(R3), is a weak solution of system (1.1).

Lemma 2.1. Suppose that h ∈ L2(R3) is a radial function and h(x) 6≡ 0. Let (A1)
and (A9) hold, then there exist r, α,Λ > 0 such that Iλ|Sr ≥ α holds for λ > 0,
|h|2 < Λ.

Proof. By (A1) and (A9), there exist L,C > 0 such that

G(t) ≤ −Lt2 + C|t|6 ∀t ∈ R. (2.3)

For each λ > 0, by the Hölder and Sobolev inequalities,

Iλ(u) ≥ 1

2

∫
R3

|∇u|2 dx+ L

∫
R3

u2 dx− C
∫
R3

|u|6 dx− |h|2|u|2

≥ min
{1

2
, L
}
‖u‖2 − CS−3‖u‖6 − |h|2‖u‖

= ‖u‖
(

min
{1

2
, L
}
‖u‖ − CS−3‖u‖5 − |h|2

)
.

(2.4)



EJDE-2021/01 NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS 5

Setting f(t) = min{ 1
2 , L}t− CS

−3t5 for t ≥ 0, there exists

r =
(min{ 1

2 , L}
5CS−3

)1/4

> 0

such that

max
t≥0

f(t) = f(r) =
4(min{ 1

2 , L})
5/4

5(5CS−3)1/4
= Λ.

Hence from (2.4) we deduce that if |h|2 < Λ, there exists α = Λ−|h|2 > 0 satisfying
Iλ|Sr ≥ α for all λ > 0. This completes the proof. �

As in [6], we set

g1(t) =

{
(g(t) +mt)+, t ≥ 0,

(g(t) +mt)−, t ≤ 0,

g2(t) = g1(t)− g(t)−mt for t ∈ R. (2.5)

Clearly, g1 and g2 satisfy

lim
t→0

g1(t)

t
= lim
|t|→∞

g1(t)

|t|4t
= 0, (2.6)

g2(t)t ≥ 0 for all t ∈ R. (2.7)

Lemma 2.2. Suppose that (A1) and (A9) hold. Let {un} ⊂ H1
r (R3) be a bounded

Palais-Smale sequence of Iλ, then {un} has a convergent subsequence in H1
r (R3).

Proof. Since {un} is bounded, up to a subsequence, there exists u ∈ H1
r (R3) such

that

un ⇀ u in H1
r (R3),

un → u in Ls(R3), s ∈ (2, 6),

un → u a.e. in R3.

We now show that un → u in H1
r (R3). We recall that {un} is a bounded Palais-

Smale sequence for Iλ, namely, {Iλ(un)} is bounded and I ′λ(un) → 0. Combining
this with (2.5), we have

〈I ′λ(un)− I ′λ(u), un − u〉

=

∫
R3

|∇(un − u)|2 +m(un − u)2 dx+ λ

∫
R3

(φunun − φuu)(un − u) dx

−
∫
R3

(g1(un)− g1(u))(un − u) dx+

∫
R3

(g2(un)− g2(u))(un − u) dx

≥ min{1,m}‖un − u‖2 + λ

∫
R3

(φunun − φuu)(un − u) dx

−
∫
R3

(g1(un)− g1(u))(un − u) dx

+

∫
R3

g2(un)un − g2(un)u− g2(u)(un − u) dx.

(2.8)

It is clear that

〈I ′λ(un)− I ′λ(u), un − u〉 → 0 as n→∞. (2.9)



6 L.-X. HUANG, X.-P. WU, C.-L. TANG EJDE-2021/01

When n→∞, by the Hölder and Sobolev inequalities, we easily obtain∫
R3

(φunun − φuu)(un − u) dx ≤ (|φun |6|un|12/5 + |φu|6|u|12/5)|un − u|12/5 → 0.

(2.10)
Combining Strauss’s lemma with (2.6) (see [6, Theorem A.I]), we have∫

R3

(g1(un)− g1(u))(un − u) dx→ 0 as n→∞. (2.11)

From (2.7) and Fatou’s lemma, one deduces that

lim inf
n→∞

∫
R3

g2(un)un dx ≥
∫
R3

g2(u)u dx.

Furthermore, one obtains

lim
n→∞

∫
R3

g2(un)un − g2(un)u− g2(u)(un − u) dx ≥ 0 as n→∞. (2.12)

Using (2.9)–(2.12) in (2.8), we conclude that un → u in H1
r (R3) as n → ∞. This

completes the proof. �

Theorem 2.3. Suppose that h ∈ L2(R3) is a radial function and h(x) 6≡ 0. Let (A1)
and (A9) hold, then (1.1) has a nontrivial solution u1 ∈ H1

r (R3) with Iλ(u1) < 0
for λ > 0, |h|2 < Λ, where Λ is given by Lemma 2.1.

Proof. We choose a ϕ ∈ H1
r (R3) such that

∫
R3 h(x)ϕ(x) dx > 0. Moreover, by (A1)

and (A9), for any δ > 0, there exists Cδ > 0 such that

|G(t)| ≤ Cδ|t|2 + δ|t|6 for all t ∈ R. (2.13)

Hence, for t > 0 small enough, we have

Iλ(tϕ) ≤
∫
R3

t2

2
|∇ϕ|2 +

λt4

4
φϕϕ

2 dx+

∫
R3

Cδt
2|ϕ|2 + δt6|ϕ|6 dx−

∫
R3

thϕ dx < 0.

Thus, we obtain c0 = infu∈B̄r Iλ(u) < 0, where r is given by Lemma 2.1. Fur-
thermore, by Ekeland’s variational principle [12], there is a minimizing sequence
{un} ⊂ B̄r of c0 such that

c0 ≤ Iλ(un) ≤ c0 +
1

n
,

Iλ(ϕ) ≥ Iλ(un)− 1

n
‖ϕ− un‖ for all ϕ ∈ B̄r.

It is standard to show that {un} is a bounded (PS)c0 sequence of Iλ. By Lemma
2.2 we prove that {un} possesses a convergent subsequence. Thus, we conclude
that there exists u1 ∈ H1

r (R3) such that Iλ(u1) = c0 < 0 and I ′λ(u1) = 0. So we
completed the proof. �

3. A weak solution with positive energy

Following [2], we introduce a cut-off function χ ∈ C∞(R+, [0, 1]) satisfying

χ(t) = 1, t ∈ [0, 1],

0 ≤ χ(t) ≤ 1, t ∈ (1, 2),

χ(t) = 0, t ≥ 2,

|χ′|∞ ≤ 2



EJDE-2021/01 NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS 7

and the truncated functional Iλ,T : H1
r (R3)→ R is defined as

Iλ,T (u)

=
1

2

∫
R3

|∇u|2 dx+
λ

4
hT (u)

∫
R3

φuu
2 dx−

∫
R3

G(u) dx−
∫
R3

h(x)u dx

=
1

2

∫
R3

|∇u|2 +mu2 dx+
λ

4
hT (u)

∫
R3

φuu
2 dx−

∫
R3

G1(u)−G2(u) + h(x)u dx,

where T > 0, hT (u) = χ(T−2‖u‖2). It is standard to prove that Iλ,T is of class C1

whose derivative is given by

〈I ′λ,T (u), v〉

=

∫
R3

∇u · ∇v +muv dx+ λhT (u)

∫
R3

φuuv dx

+
aλ,T (u)

2

∫
R3

∇u · ∇v + uv dx−
∫
R3

g1(u)v − g2(u)v dx−
∫
R3

hv dx,

(3.1)

for all v ∈ H1
r (R3), where

aλ,T (u) = λT−2χ′(T−2‖u‖2)

∫
R3

φuu
2 dx. (3.2)

For T sufficiently large and λ sufficiently small, we can find a critical point u such
that ‖u‖ ≤ T and prove that u is a critical point of Iλ.

We shall use the following Pohožaev type identity. The proof can be done simi-
larly to that in [2] and details are omitted here.

Lemma 3.1. Suppose that h ∈ L2(R3) is a radial function and h(x) 6≡ 0. Let (A1),
(A9), (A10) hold and u ∈ H1

r (R3) is a weak solution of (1.1), then the following
Pohožaev type identity holds

Pλ(u) :=
1

2

∫
R3

|∇u|2 dx+
5λ

4

∫
R3

φuu
2 dx−

∫
R3

3G(u) + 3hu+ (x · ∇h)u dx = 0.

Lemma 3.2. For 8λT̃ < min{1,m}, every bounded Palais-Smale sequence of Iλ,T
admits a convergent subsequence in H1

r (R3), where T̃ = a1T
2, a1 is given by (2.2).

Proof. Let {un} be a bounded Palais-Smale sequence of Iλ,T . Repeating the proof
of Lemma 2.2, we easily obtain

on(1) = 〈I ′λ,T (un)− I ′λ,T (u), un − u〉
≥ min{1,m}〈un, un − u〉 −max{1,m}〈u, un − u〉

+ λhT (un)

∫
R3

φunun(un − u) dx− λhT (u)

∫
R3

φuu(un − u) dx

+
aλ,T (un)

2
〈un, un − u〉 −

aλ,T (u)

2
〈u, un − u〉

−
∫
R3

(g1(un)− g1(u))(un − u) dx+

∫
R3

(g2(un)− g2(u))(un − u) dx

=
(

min{1,m}+
aλ,T (un)

2

)
〈un, un − u〉,



8 L.-X. HUANG, X.-P. WU, C.-L. TANG EJDE-2021/01

this shows that
(

min{1,m} +
aλ,T (un)

2

)
〈un, un − u〉 → 0 as n → ∞. By (2.2) and

(3.2), we have

|aλ,T (un)| ≤ λT−2|χ′(T−2‖un‖2)|
∣∣ ∫

R3

φunu
2
n dx

∣∣ < 8λT̃ . (3.3)

For 8λT̃ < min{1,m}, we conclude that

min{1,m}+
aλ,T (un)

2
≥ min{1,m} − 4λT̃ > 0.

Then 〈un, un − u〉 → 0. Combining with un ⇀ u in H1
r (R3), this implies that

un → u in H1
r (R3). The proof is complete. �

Next, we prove that the functional Iλ,T possesses a mountain pass geometry.

Lemma 3.3. Suppose that h ∈ L2(R3) is a radial function and h(x) 6≡ 0. Let (A1),
(A3), (A9) hold, then there exist r, α,Λ > 0 such that for λ > 0 and |h|2 < Λ, we
have

(i) Iλ,T |Sr ≥ α,
(ii) there exists a function ω ∈ H1

r (R3)\{0} such that ‖ω‖ > r and Iλ,T (ω) < 0.

Proof. (i) Repeating the proof of Lemma 2.1, we prove the statement holds.
(ii) For any given R > 1, we define

ωR(x) =


ζ, |x| ≤ R,
ζ(R+ 1− |x|), R < |x| ≤ R+ 1,

0, |x| > R+ 1.

Through a direct calculation, we conclude that∫
R3

|∇ωR|2 dx = ζ2 meas{BR+1 −BR},∫
R3

G(ωR) dx ≥ G(ζ) meas{BR} −meas{BR+1 −BR}( max
s∈[0,ζ]

|G(s)|),∫
R3

|h(x)ωR| dx ≤ Λζ(meas{BR+1})1/2,

where meas{·} denotes Lebesgue measure. Then there exist some Ci > 0 (i =
1, 2, 3, 4, 5) such that ∫

R3

|∇ωR|2 dx ≤ C1R
2, (3.4)∫

R3

G(ωR) dx ≥ C2R
3 − C3R

2, (3.5)∫
R3

|h(x)ωR| dx ≤ C4(R+ 1)3/2. (3.6)



EJDE-2021/01 NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS 9

Defining ωR,θ : ωR(xθ ) for θ > 0 and combining with (3.4)–(3.6), we obtain

Iλ,T (ωR,θ) =
θ

2

∫
R3

|∇ωR|2 dx+
λ

4
hT (ωR,θ)

∫
R3

φωR,θω
2
R,θ dx

− θ3

∫
R3

G(ωR) dx−
∫
R3

hωR,θ dx

≤ θ

2
C1R

2 − θ3(C2R
3 − C3R

2) + θ3/2C4(R+ 1)3/2

+
λ

4
χ
(θ ∫R3 |∇ωR|2 dx+ θ3

∫
R3 ω

2
R dx

T 2

)∫
R3

φωR,θω
2
R,θ dx.

(3.7)

Therefore, we can choose R > 1 and θ > 0 sufficiently large such that ‖ωR,θ‖ >
max{r,

√
2T} and Iλ,T (ωR,θ) < 0. Namely, (ii) holds. This completes the proof. �

Set θ > 0 and ω̄R = ωR(·/θ). Define

γ(t) =

{
0, t = 0,

ω̄R(·/t), 0 < t ≤ 1.

It is easy to see that γ is a continuous path from 0 to ω̄R. Then, by Lemma 3.3,
we define the mountain pass level

c = inf
γ∈Γ

sup
t∈[0,1]

Iλ,T (γ(t)) > 0,

where the set of paths Γ :=
{
γ ∈ C([0, 1], H1

r (R3)) : γ(0) = 0 and Iλ,T (γ(1)) < 0
}

.

Lemma 3.4. Suppose that h ∈ L2(R3) is a radial function and h(x) 6≡ 0. Let
(A1), (A3), (A9), (A10) hold. Then there exists {un} satisfying Iλ,T (un) → c,
Pλ,T (un)→ 0, and I ′λ,T (un)→ 0 in [H1

r (R3)]∗, where

Pλ,T (un) =
(1

2
+
aλ,T (un)

4

)∫
R3

|∇un|2 dx+
3aλ,T (un)

4

∫
R3

u2
n dx

+
5λ

4
hT (un)

∫
R3

φunu
2
n dx

−
∫
R3

3G(un) dx+ 3hun + (x · ∇h)un dx.

(3.8)

Proof. Following Jeanjean [15], we define the map Φ : R ×H1
r (R3) → H1

r (R3) for
σ ∈ R and v ∈ H1

r (R3) by Φ(σ, v)(x) = v(e−σx). For every σ ∈ R and v ∈ H1
r (R3),

the functional Iλ,T ◦ Φ is computed as

Iλ,T (Φ(σ, v)) =
eσ

2

∫
R3

|∇v|2 dx+
λe5σ

4
hT (v(e−σx))

∫
R3

φvv
2 dx

− e3σ

∫
R3

G(v) + h(eσx)v dx.

In view of (A1), (A3), and (A9), Iλ,T ◦ Φ is continuously Fréchet-differential on
R×H1

r (R3). We also define

c̄ = inf
γ̄∈Γ̄

sup
t∈[0,1]

(Iλ,T ◦ Φ)(γ̄(t)),



10 L.-X. HUANG, X.-P. WU, C.-L. TANG EJDE-2021/01

where the class Γ̄ =
{
γ̄ ∈ C([0, 1],R×H1

r (R3)) : γ̄(0) = (0, 0), (Iλ,T ◦Φ)(γ̄(1)) < 0
}

.

Since Γ =
{

Φ ◦ γ̄ : γ̄ ∈ Γ̄
}

, we verify that c = c̄. Let γ̄ = (0, γ), for every ε ∈ (0, c2 ),
there exists γ ∈ Γ such that

sup(Iλ,T ◦ Φ)(0, γ) ≤ c+ ε.

Then, by [25, Theorem 2.8], there exists (σ, v) ∈ R×H1
r (R3) such that

(a) c− 2ε ≤ (Iλ,T ◦ Φ)(σ, v) ≤ c+ 2ε,

(b) dist{(σ, v), (0, γ)} ≤ 2
√
ε, where dist{(σ, v), (τ, ϑ)} = (|σ−τ |2+‖v−ϑ‖2)1/2,

(c) ‖(Iλ,T ◦ Φ)′(σ, v)‖ ≤ 2
√
ε.

Then there exists a sequence {(σn, vn)} ⊂ R × H1
r (R3) such that, as n → ∞, we

have

σn → 0, (Iλ,T ◦ Φ)(σn, vn)→ c, (Iλ,T ◦ Φ)′(σn, vn)→ 0 in [H1
r (R3)]∗.

It is easy to prove that, for every (h, ι) ∈ R×H1
r (R3),

(Iλ,T ◦ Φ)′(σn, vn)[h, ι] = I ′λ,T (Φ(σn, vn))[Φ(σn, ι)] + Pλ,T (Φ(σn, vn))h. (3.9)

Then, taking un = Φ(σn, vn), we obtain Iλ,T (un) → c. Further, set (h, ι) = (1, 0)
and (h, ι) = (0,Φ(−σn, ψ)) in (3.9) in order, we conclude that Pλ,T (un) → 0 and
〈I ′λ,T (un), ψ〉 → 0. As a consequence, we have

Iλ,T (un)→ c, Pλ,T (un)→ 0, I ′λ,T (un)→ 0 in [H1
r (R3)]∗. (3.10)

Thus, we complete the proof. �

Lemma 3.5. Under the assumptions of Lemma 3.4, let {un} be given by (3.10).

Then there exist T0 > 1 and λ0 > 0 satisfying 17λ0T
2
0 T̃0 < min{1,m} such that

‖un‖ ≤ T0 for λ ∈ (0, λ0).

Proof. Motivated by [28], we argue by contradiction. Suppose for every T > 1,

there exists λT > 0 satisfying 17λTT
2T̃ < min{1,m} such that

lim sup
n→∞

‖un,λT ‖ > T. (3.11)

For simplicity, we denote un,λT , λT by un, λ respectively. By (3.8) and (3.10), {un}
satisfies the identity(1

2
+
aλ,T (un)

4

)∫
R3

|∇un|2 dx+
3aλ,T (un)

4

∫
R3

u2
n dx+

5λ

4
hT (un)

∫
R3

φunu
2
n dx

=

∫
R3

3G1(un)− 3G2(un)− 3m

2
u2
n + 3hun dx+

∫
R3

(x · ∇h)un dx+ on(1).

(3.12)
Actually, since Iλ,T (un)→ c,

1

2

∫
R3

|∇un|2 +mu2
n dx+

λ

4
hT (un)

∫
R3

φunu
2
n dx

=

∫
R3

G1(un)−G2(un) dx+

∫
R3

hun dx+ c+ on(1).

(3.13)

By (A10), there exists a function ξ(x) ∈ L 6
5 (R3) such that |∇h(x)||x| ≤ ξ(x) for

any x ∈ R3. Then from (A10) and (3.12)–(3.13) it follows that∫
R3

|∇un|2 dx



EJDE-2021/01 NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS 11

≤ 3c+
3|aλ,T (un)|

4
‖un‖2 +

λ

2
hT (un)

∫
R3

φunu
2
n dx−

∫
R3

(x · ∇h)un dx+ on(1)

≤ 3c+
3|aλ,T (un)|

4
‖un‖2 +

λ

2
hT (un)

∫
R3

φunu
2
n dx+ |ξ| 6

5
|un|6 + on(1)

≤ 3c+
3|aλ,T (un)|

4
‖un‖2 +

λ

2
hT (un)

∫
R3

φunu
2
n + C

(∫
R3

|∇un|2 dx
)1/2

+ on(1),

we have ∫
R3

|∇un|2 dx− C
(∫

R3

|∇un|2 dx
)1/2

≤ 3c+
3|aλ,T (un)|

4
‖un‖2 +

λ

2
hT (un)

∫
R3

φunu
2
n dx+ on(1).

(3.14)

Next, we turn to the estimate of right-hand side of (3.14). By the definition of χ,
we obtain

λhT (un)

∫
R3

φunu
2
n dx < 4λT 2T̃ . (3.15)

By Lemma 3.3 and (3.7), there exists a2 > 0 such that

c ≤ max
θ
Iλ,T (ωR,θ(x))

≤ max
θ

{θ
2
C1R

2 − θ3(C2R
3 − C3R

2) + θ3/2C4(R+ 1)3/2
}

+ max
θ

λ

4
χ
(‖ωR,θ(x)‖2

T 2

)∫
R3

φωR,θω
2
R,θ dx

= a2 +Aλ(T ).

(3.16)

If ‖ωR,θ(x)‖2 ≥ 2T 2, then χ
(
‖ωR( xθ )‖2

T 2

)
= 0. Thus, by (3.15), Aλ(T ) ≤ λT 2T̃ . By

(3.3), one has

3|aλ,T (un)|
4

‖un‖2 < 12λT 2T̃ . (3.17)

Then, by (3.14)–(3.17) we obtain∫
R3

|∇un|2 dx− C
(∫

R3

|∇un|2 dx
)1/2

≤ 3(a2 + λT 2T̃ ) + 12λT 2T̃ + 2λT 2T̃ + on(1)

= 3a2 + 17λT 2T̃ + on(1),

which yields(∫
R3

|∇un|2 dx
)1/2

≤ C

2
+

√
C2

4
+ 3a2 + 17λT 2T̃ + on(1). (3.18)

Meanwhile, since I ′λ,T (un)→ 0, by (2.6) and (3.1),(
min{1,m}+

aλ,T (un)

2

)
‖un‖2 + λhT (un)

∫
R3

φunu
2
n dx+

∫
R3

g2(un)un dx

≤
∫
R3

g1(un)un dx+

∫
R3

hun dx+ on(1)

≤ ε|un|22 + C|un|66 + |h|2‖un‖+ on(1).

(3.19)



12 L.-X. HUANG, X.-P. WU, C.-L. TANG EJDE-2021/01

Thus, by (2.7) and (3.18)–(3.19) we obtain(
min{1,m}+

aλ,T (un)

2
− ε
)
‖un‖2 − |h|2‖un‖

≤ C|un|66 + on(1)

≤ CS−3‖un‖6D + on(1)

≤ C

S3

(C
2

+

√
C2

4
+ 3a2 + 17λT 2T̃ + on(1)

)6

+ on(1).

Since 17λT 2T̃ < min{1,m}, by (3.3) we have

min{1,m}+
aλ,T (un)

2
≥ min{1,m} − 4λT̃ >

min{1,m}
2

,

we easily get (min{1,m}
2

− ε
)
‖un‖2 − |h|2‖un‖

≤ C

S3

(C
2

+

√
C2

4
+ 3a2 + 1 + on(1)

)6

+ on(1).

(3.20)

By (3.11), (3.20) is impossible for T > 1 large enough. This completes the proof. �

Theorem 3.6. Suppose that h ∈ L2(R3) is a radial function and h(x) 6≡ 0. Let
(A1), (A3), (A9), (A10) hold. Then there exist λ0,Λ > 0 such that (1.1) has a
nontrivial solution u2 ∈ H1

r (R3) with Iλ(u2) > 0 for λ ∈ (0, λ0), |h|2 < Λ.

Proof. Combining Lemmas 3.1–3.5 and the mountain pass theorem, we can find a
critical point u2 for Iλ,T at c when λ and |h|2 are sufficiently small. By Lemma 3.5,
{un} is a (PS)c sequence of Iλ,T and satisfies ‖un‖ ≤ T , which implies that u2 is a
critical point for Iλ at c. Then we prove that there exist λ0,Λ > 0, such that (1.1)
has a nontrivial radial solution u2 with Iλ(u2) = c > 0 for λ ∈ (0, λ0), |h|2 < Λ. �

Proof of Theorem 1.2. It follows from Theorems 2.3 and 3.6, �

Proof of Corollary 1.3. For to this end, we construct a new system

−∆u+ λφu = g̃(u) + h(x) in R3,

−∆φ = u2 in R3,
(3.21)

where g̃ : R→ R is defined by

g̃(t) =

{
−mt, t ≤ 0,

g(t), t ≥ 0,

and define the energy functional Jλ : H1
r (R3) 7→ R, corresponding to system (3.21),

as

Jλ(u) =
1

2

∫
R3

|∇u|2 dx+
λ

4

∫
R3

φuu
2 dx−

∫
R3

G̃(u) dx−
∫
R3

h(x)u dx,

where G̃(t) =
∫ t

0
g̃(s)ds. It is standard to prove that Jλ is a well defined C1-

functional. Then, under the assumptions of Theorem 1.2, there exist λ̃0, Λ̃ > 0
such that system (3.21) has two nontrivial radial solutions ũ1, ũ2 for λ ∈ (0, λ̃0),



EJDE-2021/01 NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS 13

|h|2 < Λ̃, which satisfy that Jλ(ũ1) < 0 < Jλ(ũ2). Further, letting ũ−1 be a test
function, one has

〈J ′λ(ũ1), ũ−1 〉 =

∫
R3

|∇ũ−1 |2 dx+

∫
R3

φũ−
1

(ũ−1 )2 dx+

∫
R3

m|ũ−1 |2 dx−
∫
R3

hũ−1 dx,

which implies that ũ−1 = 0 from h(x) ≥ 0 in R3, so ũ1(x) ≥ 0 in R3. Namely, ũ1 is
also the nonnegative radial solution of (1.1) from the definition of g̃. By (A1) and
(A9), there exist some L̄ > 0 such that

g(t) ≥ −L̄(|t|+ |t|5) for all t ∈ R. (3.22)

It is clear that ũ1 solves the equation

−∆u+ λφu+ L̄(1 + u4)u = g(u) + L̄(u+ u5) + h(x).

From the regular estimates of elliptic equations, we may deduce that ũ1 ∈ L∞loc(R3)
and φũ1 ∈ L∞loc(R3). Therefore, there exists C(Ω) > 0 such that −∆ũ1+C(Ω)ũ1 ≥ 0
in any bounded domain Ω. Applying the strong maximum principle [14, Theorem
8.9], we derive that ũ1(x) > 0 in R3. Similarly, it can be proved that ũ2(x) > 0 in
R3. The proof is complete. �

Acknowledgments. This research was supported by the National Natural Science
Foundation of China (No.11971393).

The authors want to express their gratitude to the reviewers for careful reading
and valuable suggestions which led to an improvement of the original manuscript.

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Lan-Xin Huang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China.
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Email address: huang101908@qq.com

Xing-Ping Wu (corresponding author)

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Email address: wuxp@swu.edu.cn

Chun-Lei Tang
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Email address: tangcl@swu.edu.cn


	1. Introduction and statement of main results
	2. A weak solution with negative energy
	3. A weak solution with positive energy
	Acknowledgments

	References