Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 28, pp. 1–14.

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

MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS

SCHRÖDINGER-POISSON SYSTEM WITH p-LAPLACIAN

LANXIN HUANG, JIABAO SU

Abstract. This article concerns the existence of solutions to the Schrödinger-
Poisson system

−∆pu+ |u|p−2u+ λφu = |u|q−2u+ h(x) in R3,

−∆φ = u2 in R3,

where 4/3 < p < 12/5, p < q < p∗ = 3p/(3 − p), ∆pu = div(|∇u|p−2∇u),

λ > 0, and h 6= 0. The multiplicity results are obtained by using Ekeland’s
variational principle and the mountain pass theorem.

1. Introduction and statement of main results

This article concerns the existence of solutions to the Schrödinger-Poisson system

−∆pu+ |u|p−2u+ λφu = |u|q−2u+ h(x) in R3,

−∆φ = u2 in R3,
(1.1)

where 4/3 < p < 12/5, p < q < p∗ = 3p
3−p , ∆pu = div(|∇u|p−2∇u), λ > 0, and

h 6= 0. The system (1.1) can be viewed as a perturbation of the system

−∆pu+ |u|p−2u+ λφu = |u|q−2u in R3,

−∆φ = u2 in R3.
(1.2)

This system was first considered by Du, Su, and Wang in [11] where the variational
framework was built and the existence of nontrivial solutions was established via
the mountain pass theorem. For p = 2, the system (1.2) reduces to the following
classical Schrödinger-Poisson system

−∆u+ u+ λφu = |u|q−2u in R3,

−∆φ = u2 in R3,
(1.3)

where λ > 0 and q ∈ (2, 6). Such a system, also known as the nonlinear Schrödinger-
Maxwell equation, has an interesting physical context. According to a classical
model, the interaction of a charged particle with an electromagnetic field can be
described by coupling a nonlinear Schrödinger equation and a Poisson equation.
For more details on the physical aspects of the system we refer to the pioneering
works of Benci and Fortunado [5, 6] and the references therein. In the past decades,

2020 Mathematics Subject Classification. 35J10, 35J50, 35J60, 35J92.
Key words and phrases. Nonhomogeneous Schrödinger-Poisson system; variational methods;

multiple solutions; p-Laplacian.
©2023. This work is licensed under a CC BY 4.0 license.
Submitted July 8, 2022. Published March 11, 2023.

1



2 L. HUANG, J. SU EJDE-2023/28

the existence of solutions to the system (1.3) has been discussed in [4] for q ∈ (3, 6),
in [9, 10] for q ∈ [4, 6), and in [2, 3, 21, 25, 31] for q ∈ (2, 6) or general nonlinearity.

For p = 2, the system (1.1) reduces to the nonhomogeneous Schrödinger-Poisson
system

−∆u+ u+ λφu = |u|q−2u+ h(x) in R3,

−∆φ = u2 in R3,
(1.4)

where λ > 0, q ∈ (2, 6) and h(x) 6≡ 0. In [22], Salvatore obtained multiple ra-
dial solutions to the system (1.4) for q ∈ (4, 6) and h ∈ L2(R3) being radial
with small L2-norm. In [16], Jiang, Wang and Zhou considered the system (1.4)
with h ∈ C1(R3) ∩ L2(R3) being a nonnegative radial function and satisfying
(x,∇h) ∈ L2(R3). Applying the Ekeland’s variational principle and the moun-
tain pass theorem, it was proved in [16] that the system (1.4) admitted two radial
solutions for q ∈ (2, 6) with small L2-norm |h|L2(R3) of h and for q ∈ (2, 3] with
λ > 0 also small. For other works related to the system (1.4) or to similar systems
involving certain potentials, we refer to [8, 13, 17, 20, 26, 27, 30, 32, 33] and the
references therein.

After an accurate bibliographic review, we see that it is open question the exis-
tence of multiple solutions to the quasilinesr system (1.1) with 4/3 < p < 12/5 and
h 6= 0. Inspired by this fact, We aim to establish the existence of multiple solutions
to system (1.1). We use τ ′ = τ

τ−1 to denote the Hölder conjugate of τ > 1. We
impose on h the following assumption.

(H1) h is a nonzero radial function and for (p∗)′ ≤ s ≤ p′,
(i) h ∈ Ls(R3) with the Ls-norm denoted by |h|Ls(R3);

(ii) (x,∇h) ∈ Ls(R3) where the gradient ∇h is in the weak sense.

We will prove the following theorems.

Theorem 1.1. Assume that (H1) holds and 6p
p+2 < q < p∗. Then there exists Λ > 0

such that for |h|Ls(R3) < Λ the system (1.1) admits two solutions for any λ > 0.

Theorem 1.2. Assume that (H1)(i) holds and p < q ≤ 6p
p+2 . There exist Λ > 0

and λ∗ > 0 such that for |h|Ls(R3) < Λ, system (1.1) admits two solutions for any
λ ∈ (0, λ∗).

Remark 1.3. The first attempt of the study on the Schrödinger-Poisson system
(1.2) with p-Laplacian were made in [11]. Now the results in Theorems 1.1 and 1.2
extend the results in [16, 22] from p = 2 to the quasilinear case 4/3 < p < 12/5.
This range of p was first determined in [11]. We observe a phenomenon that the
solvability of the system (1.1) can be considered for a large class of radial functions
h satisfying (H1). In this sense the existence results in [16] may be extended to the
case that h and (x,∇h) belonging to Ls(R3) with 6/5 ≤ s ≤ 2.

Notice that for p 6= 2, it is difficult to prove the Pohožaev identity which is
essential to establish the boundedness of Palais-Smale sequences for q ∈ (3, 6) in
[16]. To overcome this difficulty, for 6p/(p+ 2) < q < p∗ we introduce an auxiliary
functional and use an indirect method to do that: see our proof of Lemma 4.3. It
also should be pointed out that our method is more applicable. As far as we know,
this article is the first attempt to study the nonhomogeneous Schrödinger-Poisson
system with p-Laplacian.

The proofs of the main results will be obtained by exploiting suitable variational
methods. In Section 2, we give some preliminary results concerning the variational



EJDE-2023/28 SCHRÖDINGER-POISSON SYSTEMS WITH p-LAPLACIAN 3

structure for the system (1.1). In Section 3, with the aid of the Ekeland’s variational
principle [12], we obtain by Theorem 3.3 a solution of (1.1) with negative energy
for p < q < p∗. In Section 4 we obtain a solution of (1.1) with positive energy
and discuss with two cases of 6p

p+2 < q < p∗ and p < q ≤ 6p
p+2 . In Subsection 4.1,

we use the scaling technique beginning in [14] and developing in [11] to obtain the
boundedness of a Palais-Smale sequence for 6p

p+2 < q < p∗ and find a positive energy

solution by using the mountain pass theorem [1], see Theorem 4.1. In Subsection
4.2, by using the cut-off technique as in [15] and combining some delicate analysis,
we prove a positive energy solution of (1.1) with p < q ≤ 6p

p+2 and λ > 0 small, see

Theorem 4.4. Then Theorems 1.1 and 1.2 will follow from Theorem 3.3, Theorems
4.1 and 4.4.

2. Preliminaries

In this section we give some preliminary results related to the variational struc-
ture of system (1.1). We will use the following function spaces.

• Ls(Ω), the Lebesgue space endowed with the norm |u|Ls(Ω) =
(∫

Ω
|u|s dx

)1/s
for

1 ≤ s <∞.

• W 1,p(R3), the Sobolev space with the norm ‖u‖ =
( ∫

R3 |∇u|p + |u|p dx
)1/p

, and

W 1,p
r (R3) =

{
u ∈W 1,p(R3) : u(x) = u(|x|)

}
.

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

R3 |∇u|2 dx
)1/2

.

It is a Hilbert space with the inner product 〈v, w〉 =
∫
R3 ∇v∇w dx.

It follows from the classical Sobolev embedding theorems that W 1,p(R3) ↪→
L`(R3) are continuous for all ` ∈ [p, p∗] and D1,2(R3) ↪→ L6(R3) is continuous.
Restricted to the radial case, it holds that the embedding W 1,p

r (R3) ↪→ L`(R3) is
compact for any p < ` < p∗. See [19, Theorem II.1] or [23, Theorem 1].

We will use C to denote various positive constants. We will use the following
elementary inequality (see [24, p240]) in later arguments: There exists cp > 0 such
that for all ξ, η ∈ R3, we have(

|ξ|p−2ξ − |η|p−2η, ξ − η
)
R3 ≥ cp|ξ − η|p for p ≥ 2,(

|ξ|+ |η|)2−p(|ξ|p−2ξ − |η|p−2η, ξ − η
)
R3 ≥ cp|ξ − η|2 for 1 < p < 2.

(2.1)

For each fixed u ∈W 1,p(R3), we define a linear functional K : D1,2(R3)→ R by

K(v) =

∫
R3

u2v dx.

By the Hölder and Sobolev inequalities, we have

|K(v)| ≤
(∫

R3

|u|12/5 dx
)5/6(∫

R3

|v|6 dx
)1/6

≤ C‖u‖2‖v‖D.

Therefore K is continuous on D1,2(R3). By the Lax-Milgram theorem, there exists
a unique φu ∈ D1,2(R3) satisfying the equation −∆φu = u2. According to [18,

Theorem 6.21], φu has the explicit expression φu(x) = 1
4π

∫
R3

u2(y)
|x−y|dy ≥ 0.

It defines a mapping u 7→ φu from W 1,p(R3) to D1,2(R3) such that φu ≥ 0 solves
uniquely the Poisson equation −∆φ = u2 for u ∈W 1,p(R3).

Proposition 2.1 ([11, Proposition 2.1]). The mapping u 7→ φu enjoys the following
properties.



4 L. HUANG, J. SU EJDE-2023/28

(i) ‖φu‖D ≤ A‖u‖2 for all u ∈W 1,p(R3) where A > 0 is a constant;
(ii) if un ⇀ u in W 1,p(R3), then φun ⇀ φu in D1,2(R3);

(iii) if u ∈W 1,p
r (R3) then φu ∈ D1,2

r (R3) := {φ ∈ D1,2(R3) : φ(x) = φ(|x|)}.

We note here that the third conclusion comes from a fact that the convolution
of two radial functions is still radial.

Now we are ready to establish the variational framework of (1.1). For h ∈ Ls(R3)
with (p∗)′ ≤ s ≤ p′, arguing as in [5, 6], by Proposition 2.1 and the implicit function
theorem, the functional

Iλ(u) =
1

p

∫
R3

(|∇u|p + |u|p) dx+
λ

4

∫
R3

φuu
2 dx− 1

q

∫
R3

|u|q dx−
∫
R3

h(x)u dx

is a well-defined C1 functional on W 1,p(R3) with derivative

〈I ′λ(u), v〉 =

∫
R3

(|∇u|p−2∇u∇v + |u|p−2uv) dx+ λ

∫
R3

φuuv dx

−
∫
R3

|u|q−2uv dx−
∫
R3

h(x)v dx, ∀u, v ∈W 1,p(R3).

Furthermore, u ∈W 1,p(R3) is a critical point of Iλ if and only if the couple (u, φu) ∈
W 1,p(R3)×D1,2(R3) is a solution of the system (1.1). Then we will prove Theorems
1.1 and 1.2 by looking for critical points of Iλ.

The following result is crucial and can be proved by applying some ideas from
Boccardo and Murat [7]. We include the proof for completeness.

Lemma 2.2. Let {un} ⊂W 1,p(R3) be bounded and satisfy I ′λ(un)→ 0 as n→∞.
Then, up to a subsequence, there exists u ∈ W 1,p(R3) such that ∇un(x) → ∇u(x)
a.e. in R3.

Proof. Since {un} is bounded in W 1,p(R3), up to a subsequence, there exists u ∈
W 1,p(R3) such that

un ⇀ u in W 1,p(R3),

un → u in L`loc(R3), p ≤ ` < p∗,

un(x)→ u(x) a.e. in R3.

(2.2)

We will prove that

∇un(x)→ ∇u(x) a.e. in R3. (2.3)

Let υ ∈ C∞0 (R3, [0, 1]) satisfy

υ
∣∣
BR

= 1 and supt υ ⊂ B2R,

where BR = {x ∈ R3 : |x| ≤ R}. Since un ⇀ u in W 1,p(R3), it follows that

(un − u)υ ⇀ 0 in W 1,p(R3). (2.4)

Then, by (2.2) and the Hölder inequality, as n→∞,∫
R3

(|un|`−2un − |u|`−2u) [(un − u)υ] dx = o(1),∫
R3

[
(|∇un|p−2∇un − |∇u|p−2∇u)∇υ

]
(un − u) dx = o(1).

(2.5)



EJDE-2023/28 SCHRÖDINGER-POISSON SYSTEMS WITH p-LAPLACIAN 5

Using the Hölder and Sobolev inequalities, we deduce by Proposition 2.1(i) and
(2.2) that∫

R3

(φunun − φuu)(un − u)υ dx

≤ |φun |L6(B2R)|un(un − u)υ|L6/5(B2R) + |φu|L6(B2R)|u(un − u)υ|L6/5(B2R)

≤ C‖φun‖D|un(un − u)υ|L6/5(B2R) + C‖φu‖D|u(un − u)υ|L6/5(B2R)

≤ C‖un‖2|un(un − u)υ|L6/5(B2R) + C‖u‖2|u(un − u)υ|L6/5(B2R)

≤ C
(
‖un‖2|un|L12/5(B2R) + ‖u‖2|u|L12/5(B2R)

)
|un − u|L12/5(B2R)

= o(1).

(2.6)

By (2.4) and I ′λ(un)→ 0 in [W 1,p(R3)]∗, we have that as n→∞,

〈I ′λ(un)− I ′λ(u), (un − u)υ〉 = o(1). (2.7)

It follows from (2.5)–(2.7) that as n→∞,∫
R3

(
|∇un|p−2∇un − |∇u|p−2∇u

)
(∇un −∇u)υ dx = o(1). (2.8)

Set en := (|∇un|p−2∇un − |∇u|p−2∇u,∇un −∇u)R3 . Then, as n→∞,∫
BR

en dx = o(1). (2.9)

By (2.1) and (2.9), for 2 ≤ p < 12/5, we have

C

∫
BR

|∇un −∇u|p dx ≤
∫
BR

en dx = o(1), (2.10)

and for 4/3 < p < 2,

C

∫
BR

|∇un −∇u|p dx ≤
∫
BR

ep/2n (|∇un|+ |∇u|)
p(2−p)

2 dx

≤
(∫

BR

en dx
)p/2(∫

BR

(|∇un|+ |∇u|)p dx
) 2−p

2

≤ C
(∫

BR

en dx
)p/2

.

(2.11)

It follows from (2.9)–(2.11) that

lim
n→∞

∫
BR

|∇un −∇u|p dx = 0.

Up to a subsequence, we have ∇un(x) → ∇u(x) a.e. in BR. It follows from the
arbitrariness of BR that (2.3) holds. The proof is complete. �

3. A solution with negative energy

In this section we find a solution of (1.1) with negative energy for p < q < p∗,
and h satisfying (H1)(i) and small |h|Ls(R3).

Lemma 3.1. Assume that h ∈ Ls(R3) with (p∗)′ ≤ s ≤ p′. Then there exist ρ > 0,
Λ > 0 and α > 0 such that Iλ(u) ≥ α for u ∈ W 1,p(R3) with ‖u‖ = ρ, λ > 0 and
|h|Ls(R3) < Λ.



6 L. HUANG, J. SU EJDE-2023/28

Proof. For u ∈ W 1,p(R3) and λ > 0, since φu ≥ 0, it follows from Hölder and
Sobolev inequalities that

Iλ(u) ≥ 1

p
‖u‖p − 1

q
|u|qLq(R3) − |h|Ls(R3)|u|Ls′ (R3)

≥ 1

p
‖u‖p −

Sqq
q
‖u‖q − Ss′ |h|Ls(R3)‖u‖

= ‖u‖
(1

p
‖u‖p−1 −

Sqq
q
‖u‖q−1 − Ss′ |h|Ls(R3)

)
,

(3.1)

where S` denotes the embedding constant of W 1,p(R3) ↪→ L`(R3) for p ≤ ` ≤ p∗.

Since q > p, there exists a unique ρ > 0 such that the function f(t) = 1
p t
p−1−Sqq

q t
q−1

attains its unique maximum f(ρ) = maxt≥0 f(t) > 0. Take Λ = f(ρ)/Ss′ and
α = ρ

(
f(ρ)− Ss′ |h|Ls(R3)

)
. Then by (3.1) we have that when |h|Ls(R3) < Λ,

Iλ(u) ≥ α for any ‖u‖ = ρ. �

Next we work on the Sobolev space W 1,p
r (R3) of radial functions.

Lemma 3.2. Assume that h satisfies (H1)(i). Then each bounded sequence {un} ⊂
W 1,p
r (R3) satisfying I ′λ(un)→ 0 has a strongly convergent subsequence.

Proof. Let {un} ⊂ W 1,p
r (R3) be bounded. Going if necessary to a subsequence,

there exists u ∈W 1,p
r (R3) such that

un ⇀ u in W 1,p
r (R3),

un → u in Lq(R3), p < q < p∗,

un(x)→ u(x) a.e. in R3.

(3.2)

We will complete the proof by showing un → u in W 1,p
r (R3). By I ′λ(un) → 0 and

(3.2) we obtain
〈I ′λ(un)− I ′λ(u), un − u〉 → 0, as n→∞. (3.3)

By Proposition 2.1, the boundedness of {un}, the Hölder inequality and (3.2), we
obtain that, as n→∞,∫

R3

(φunun − φuu)(un − u) dx = o(1),∫
R3

(|un|q−2un − |u|q−2u)(un − u) dx = o(1).

(3.4)

It follows from (3.3) and (3.4) that∫
R3

en + (|un|p−2un − |u|p−2u)(un − u) dx = o(1). (3.5)

For 2 ≤ p < 12/5, by (2.1) we obtain∫
R3

en dx ≥ C
∫
R3

|∇un −∇u|p dx,∫
R3

(|un|p−2un − |u|p−2u)(un − u) dx ≥ C
∫
R3

|un − u|p dx.
(3.6)

For 4/3 < p < 2, from the boundedness of {un} and the proof of (2.11), we obtain∫
R3

|∇(un − u)|p dx ≤ C
(∫

R3

en dx
)p/2

, (3.7)



EJDE-2023/28 SCHRÖDINGER-POISSON SYSTEMS WITH p-LAPLACIAN 7∫
R3

|un − u|p dx ≤ C
(∫

R3

(|un|p−2un − |u|p−2u)(un − u) dx

)p/2
. (3.8)

It follows from (3.5), (3.6)–(3.8) that ‖un − u‖ → 0 as n→∞. �

Theorem 3.3. Assume that (H1)(i) holds and p < q < p∗. Then Iλ has a critical
point u∗ ∈ W 1,p

r (R3) with Iλ(u∗) < 0 for λ > 0 provided |h|Ls(R3) < Λ, where Λ
was given in Lemma 3.1.

Proof. We first find a function w ∈ W 1,p
r (R3) such that

∫
R3 h(x)w(x) dx > 0. It

follows from h ∈ Ls(R3) that |h|s−2h ∈ Ls′(R3). Then there exists a radial sequence

{hn} ⊂ C∞0 (R3) such that hn → |h|s−2h strongly in Ls
′
(R3) since C∞0 (R3) is dense

in Ls
′
(R3) and h is radial. Therefore, there exists n0 ∈ N such that∣∣hn0

− |h|s−2h
∣∣
Ls′ (R3)

≤ 1

2
|h|s−1

Ls(R3).

By Hölder’s inequality, we conclude that∫
R3

h(x)hn0
(x) dx ≥ −|h|Ls(R3)

∣∣hn0
− |h|s−2h

∣∣
Ls′ (R3)

+ |h|sLs(R3) > 0.

It is clear that hn0 ∈W 1,p
r (R3). Taking w(x) = hn0(x), we get

∫
R3 h(x)w(x) dx > 0.

Now for t > 0 small enough, we have

Iλ(tw) =
tp

p
‖w‖p +

t4

4
λ

∫
R3

φww
2 dx− tq

q

∫
R3

|w|q dx− t
∫
R3

hw dx < 0.

It follows that

c∗ = inf
u∈B̄ρ

Iλ(u) < 0,

where B̄ρ = {u ∈W 1,p
r (R3) : ‖u‖ ≤ ρ} and ρ is given by Lemma 3.1. Applying the

Ekeland variational principle [12], we obtain a sequence {un} ⊂ B̄ρ satisfying

c∗ ≤ Iλ(un) ≤ c∗ +
1

n
, (3.9)

Iλ(v) ≥ Iλ(un)− 1

n
‖v − un‖ for all v ∈ B̄ρ. (3.10)

It must be that ‖un‖ < ρ for all n ∈ N large. Otherwise, we may assume that
‖un‖ = ρ, up to a subsequence. By Lemma 3.1, we see that Iλ(un) ≥ α > 0. Then
there is a contradiction by taking the limit in (3.9) as n→∞. We can assume that
‖un‖ < ρ for all n ∈ N. Now we show that I ′λ(un)→ 0. For any z ∈W 1,p

r (R3) with
‖z‖ = 1, we choose sufficiently small δ > 0 such that ‖un + tz‖ < ρ for all |t| < δ.
By (3.10), we have

Iλ(un + tz)− Iλ(un)

t
≥ − 1

n
.

Letting t → 0, we obtain 〈I ′λ(un), z〉 ≥ −1/n. Similarly, replacing z with −z in
the above arguments, we obtain 〈I ′λ(un), z〉 ≤ 1/n. Then, we deduce that, for any
z ∈ W 1,p

r (R3) with ‖z‖ = 1, 〈I ′λ(un), z〉 → 0 as n → ∞. Thus {un} is a bounded
(PS)c∗ sequence of Iλ. Finally, by Lemma 3.2, there exists u∗ ∈ W 1,p

r (R3) such
that Iλ(u∗) = c∗ < 0 and I ′λ(u∗) = 0. �



8 L. HUANG, J. SU EJDE-2023/28

4. A solution with positive energy

In this section we find a solution of (1.1) with positive energy. In Subsection
4.1 we consider the case 6p

p+2 < q < p∗ and in Subsection 4.2 we consider the case

p < q ≤ 6p
p+2 . We still work on W 1,p

r (R3).

4.1. Case 6p
p+2 < q < p∗. In this subsection we will prove the following theorem.

Theorem 4.1. Assume that (H1) holds and 6p
p+2 < q < p∗. Then Iλ has a critical

point u∗ ∈ W 1,p
r (R3) with Iλ(u∗) > 0 for λ > 0 provided |h|Ls(R3) < Λ, where Λ

was given in Lemma 3.1.

Lemma 4.2. Assume that (H1)(i) holds and 6p
p+2 < q < p∗.

(i) There exist ρ > 0, Λ > 0 and α > 0, such that Iλ(u) ≥ α for u ∈W 1,p
r (R3)

with ‖u‖ = ρ, λ > 0, and |h|Ls(R3) < Λ.

(ii) There exists υ ∈W 1,p
r (R3)\{0} such that ‖υ‖ > ρ and Iλ(υ) < 0.

Proof. Item (i) follows from the argument of the proof of Lemma 3.1.

(ii) Take any fixed u ∈ W 1,p
r (R3)\{0} and define ut(x) = t

p+2
4−pu(tx). Then we

have

Iλ(ut) =
tβ1

p

∫
R3

|∇u|p dx+
tβ2

p

∫
R3

|u|p dx+
tβ1

4
λ

∫
R3

φuu
2 dx

− tβ3

q

∫
R3

|u|q dx− tβ4

∫
R3

h
(x
t

)
u dx,

where

β1 =
9p− 12

4− p
, β2 =

p2 + 5p− 12

4− p
,

β3 =
(p+ 2)q − 12 + 3p

4− p
, β4 =

4p− 10

4− p
.

(4.1)

It follows from 4/3 < p < 12/5 and 6p
p+2 < q that β3 > β1 > β2, β3 > 0 and β4 < 0.

Therefore there exists t0 > 0 such that Iλ(ut0) < 0. The conclusion (ii) follows by
taking υ = ut0 . �

Since Iλ(0) = 0, by Lemma 4.2, the functional Iλ satisfies the hypotheses of the
mountain pass theorem [1] and a mountain pass level of Iλ can be defined as

c = inf
γ∈Γ

max
t∈[0,1]

Iλ(γ(t)) > 0, (4.2)

where Γ = {γ ∈ C([0, 1],W 1,p
r (R3)) : γ(0) = 0 and Iλ(γ(1)) < 0}. We define an

auxiliary functional Jλ : W 1,p
r (R3) → R as follows with the numbers βi given by

(4.1):

Jλ(u) =
β1

p

∫
R3

|∇u|p dx+
β2

p

∫
R3

|u|p dx+
λβ1

4

∫
R3

φuu
2 dx− β3

q

∫
R3

|u|q dx

− β4

∫
R3

hu dx+

∫
R3

(x,∇h(x))u dx.

Lemma 4.3. Assume that (H1) holds and 6p
p+2 < q < p∗. There exists a bounded

sequence {un} ⊂W 1,p
r (R3) satisfying

Iλ(un)→ c, I ′λ(un)→ 0, Jλ(un)→ 0.



EJDE-2023/28 SCHRÖDINGER-POISSON SYSTEMS WITH p-LAPLACIAN 9

Proof. We follow the idea in Jeanjean [14] and modified in [11]. Define the map

Φ(σ, v)(x) = e
p+2
4−pσv(eσx), σ ∈ R, v ∈W 1,p

r (R3).

A simple computation shows that

Iλ(Φ(σ, v)) =
eβ1σ

p

∫
R3

|∇v|p dx+
eβ2σ

p

∫
R3

|v|p dx+
λeβ1σ

4

∫
R3

φvv
2 dx

− eβ3σ

q

∫
R3

|v|q dx− eβ4σ

∫
R3

h
( x
eσ

)
v dx,

and Iλ(Φ(0, 0)) = 0. It is standard to verify that Iλ ◦ Φ is continuously Fréchet-
differentiable on R×W 1,p

r (R3). We set

Γ̄ =
{
γ̄ ∈ C([0, 1],R×W 1,p

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

c̄ = inf
γ̄∈Γ̄

sup
t∈[0,1]

(Iλ ◦ Φ)(γ̄(t)). (4.3)

It can be proved that Γ = {Φ ◦ γ̄ : γ̄ ∈ Γ̄}. It follows that c = c̄. Let γ̄ = (0, γ).
For each ε ∈ (0, c2 ), there exists γ ∈ Γ such that

sup(Iλ ◦ Φ)(0, γ) ≤ c+ ε.

Then, by [28, Theorem 2.8], there exists (σ, v) ∈ R×W 1,p
r (R3) such that

(a) c− 2ε ≤ (Iλ ◦ Φ)(σ, v) ≤ c+ 2ε,
(b) dist{(σ, v), (0, γ)} ≤ 2

√
ε, where dist{(σ, v), (ς, w)} = (|σ − ς|2 + ‖v −

w‖2)1/2,
(c) (Iλ ◦ Φ)′(σ, v)→ 0 in [R×W 1,p

r (R3)]∗.

Therefore, there exists a sequence {(σn, vn)} ⊂ R×W 1,p
r (R3) such that as n→∞,

σn → 0, (Iλ ◦ Φ)(σn, vn)→ c, (Iλ ◦ Φ)′(σn, vn)→ 0.

For every (ζ, w) ∈ R×W 1,p
r (R3), it holds

〈(Iλ ◦ Φ)′(σn, vn), (ζ, w)〉 = 〈I ′λ(Φ(σn, vn)),Φ(σn, w)〉+ Jλ(Φ(σn, vn))ζ.

Taking un = Φ(σn, vn), we have

Iλ(un)→ c, I ′λ(un)→ 0, Jλ(un)→ 0. (4.4)

Now we prove that {un} is bounded in W 1,p
r (R3). By (4.4), for n large enough,

c+ 1 ≥ Iλ(un)− 1

β3
Jλ(un)

=
β3 − β1

pβ3

∫
R3

|∇un|p dx+
β3 − β2

pβ3

∫
R3

|un|p dx+
λ(β3 − β1)

4β3

∫
R3

φunu
2
n dx

− β3 − β4

β3

∫
R3

hun dx−
1

β3

∫
R3

(x,∇h)un dx

≥ β3 − β1

pβ3

∫
R3

|∇un|p dx+
β3 − β2

pβ3

∫
R3

|un|p dx

− β3 − β4

β3

∫
R3

hun dx−
1

β3

∫
R3

(x,∇h)un dx.

It follows that

c+ 1 +
β3 − β4

β3

∫
R3

hun dx+
1

β3

∫
R3

(x,∇h)un dx ≥
β3 − β1

pβ3
‖un‖p. (4.5)



10 L. HUANG, J. SU EJDE-2023/28

It is easy to see that
∫
R3 hun dx ≤ C‖un‖. We deduce from (H1), the Hölder and

Sobolev inequalities that∣∣ ∫
R3

(x,∇h)un dx
∣∣ ≤ (∫

R3

|(x,∇h)|s dx
)1/s(∫

R3

|un|s
′
dx
)1/s′

≤ C‖un‖.

Therefore by (4.5) that {un} is bounded in W 1,p
r (R3). �

Proof of Theorem 4.1. It follows from Lemmas 4.2, 4.3, and 3.2. �

4.2. Case p < q ≤ 6p
p+2 .

Theorem 4.4. Assume that (H1)(i) holds and p < q ≤ 6p
p+2 . Then there exists

λ∗ > 0 such that Iλ has a critical point u∗ ∈ W 1,p
r (R3) with Iλ(u∗) > 0 for each

λ ∈ (0, λ∗) provided |h|Ls(R3) < Λ, where Λ is given in Lemma 3.1.

We adopt some techniques from [15] to do the proof. We introduce a smooth
function χ ∈ C∞(R+, [0, 1]) which satisfies

χ(t) =


1 for t ∈ [0, 1

2 ],

0 for t ≥ 1,

∈ [0, 1] for t ∈ ( 1
2 , 1),

|χ′|∞ ≤ 4.

We define a penalized functional Iλ,M : W 1,p
r (R3)→ R as

Iλ,M (u) =
1

p

∫
R3

(|∇u|p + |u|p) dx+
λ

4
LM (u)

∫
R3

φuu
2 dx

− 1

q

∫
R3

|u|q dx−
∫
R3

hu dx,

(4.6)

where M > 0 and LM (u) = χ
(‖u‖p
Mp

)
. It is standard to prove that Iλ,M belongs to

C1, and for all u, v ∈W 1,p
r (R3),

〈I ′λ,M (u), v〉 = (1 + aλ,M (u))

∫
R3

(|∇u|p−2∇u∇v + |u|p−2uv) dx

+ λLM (u)

∫
R3

φuuv dx−
∫
R3

|u|q−2uv dx−
∫
R3

hv dx,

(4.7)

where

aλ,M (u) =
pλ

4Mp
χ′
(‖u‖p
Mp

)∫
R3

φuu
2 dx. (4.8)

From the definition one sees that if u is a critical point of Iλ,M and ‖u‖ ≤ M/2,
then u is a critical point of Iλ. We first verify that the penalized functional Iλ,M
possesses a mountain pass geometry for each M > 0.

Lemma 4.5. Assume that (H1)(i) holds and p < q ≤ 6p
p+2 . For every M > 0,

(i) there exist ρ > 0, Λ > 0 and α > 0, such that Iλ,M (u) ≥ α for u ∈W 1,p
r (R3)

with ‖u‖ = ρ, |h|Ls(R3) < Λ and λ > 0.

(ii) there exists ω ∈W 1,p
r (R3)\{0} such that ‖ω‖ > ρ and Iλ,M (ω) < 0.

Proof. Item (i) follows from an argument similar to the one in the proof of Lemma
3.1.



EJDE-2023/28 SCHRÖDINGER-POISSON SYSTEMS WITH p-LAPLACIAN 11

(ii) Arguing as in the proof of Theorem 3.3, we can choose a function ω1 ∈
W 1,p
r (R3) such that ‖ω1‖ = 1 and

∫
R3 h(x)ω1(x) dx > 0. For each M > 0 and

t ≥M , it follows from the definition of χ that LM (tω1) = 0. Thus

Iλ,M (tω1) =
1

p
tp − 1

q
tq
∫
R3

|ω1|q dx− t
∫
R3

h(x)ω1 dx.

Since p < q, we take ω = tMω1 and tM > M large, so that ‖ω‖ > ρ and Iλ,M (ω) < 0.
This completes the proof. �

Lemma 4.6. For M > 0 and λ > 0 fixed, each bounded sequence {un} ⊂W 1,p
r (R3)

satisfying I ′λ,M (un)→ 0 admits a strongly convergent subsequence.

Proof. Let {un} be bounded in W 1,p
r (R3). Up to a subsequence, there exists u ∈

W 1,p
r (R3) such that un ⇀ u in W 1,p

r (R3), un → u in Lq(R3) for all p < q < p∗ and
un(x)→ u(x) a.e. in R3. Therefore

〈I ′λ,M (un)− I ′λ,M (u), un − u〉 → 0, as n→∞. (4.9)

Similar to (3.4), we conclude that, as n→∞,∫
R3

φunun(un − u) dx = o(1),

∫
R3

φuu(un − u) dx = o(1),∫
R3

(|un|q−2un − |u|q−2u)(un − u) dx = o(1).

(4.10)

We set

[u, v] =

∫
R3

(|∇u|p−2∇u∇v + |u|p−2uv) dx.

From (4.7), (4.9), and (4.10), a direct computation shows that, as n→∞,

(1 + aλ,M (un)) ([un, un − u]− [u, un − u])

+ (aλ,M (un)− aλ,M (u))[u, un − u] = o(1).
(4.11)

By Proposition 2.1(i), we have that for all n ∈ N,∫
R3

φunu
2
n dx = −

∫
R3

φun∆φun dx = ‖φun‖2D ≤ A2‖un‖4. (4.12)

Notice that if ‖un‖ ≥M then χ′
(‖un‖p
Mp

)
= 0. It follows from (4.8) and (4.12) that

|aλ,M (un)| ≤ pλ

4Mp

∣∣χ′(‖un‖p
Mp

)∣∣ ∣∣ ∫
R3

φunu
2
n dx

∣∣ ≤ pλA2M4−p. (4.13)

It can be shown in a same way that |aλ,M (u)| is bounded. By Lemma 2.2, we
have that ∇un(x)→ ∇u(x) a.e. in R3. Combing with un(x)→ u(x) a.e. in R3, we
deduce by [29, Proposition 5.4.7] that

[u, un − u] = o(1). (4.14)

It follows from (4.11) and (4.14) that

[un, un − u]− [u, un − u] = o(1). (4.15)

Arguing as in the proof of Lemma 3.2 we obtain that ‖un− u‖ → 0 as n→∞. �



12 L. HUANG, J. SU EJDE-2023/28

By Lemma 4.5, we can define the following mountain pass level of Iλ,M for each
M > 0,

cM = inf
γ∈ΓM

sup
t∈[0,1]

Iλ,M (γ(t)) > 0,

where ΓM :=
{
γ ∈ C([0, 1],W 1,p

r (R3)) : γ(0) = 0, Iλ,M (γ(1)) < 0
}

. Then, by the

mountain pass theorem [1], there exists {un} ⊂W 1,p
r (R3) such that

Iλ,M (un)→ cM , I ′λ,M (un)→ 0 in [W 1,p
r (R3)]∗. (4.16)

Next we prove that {un} is bounded in W 1,p
r (R3) for large M and small λ.

Lemma 4.7. There exist M > 0 and λ∗ > 0 such that for all λ ∈ (0, λ∗), the
sequence {un} given by (4.16) satisfies

‖un‖ ≤
M

2
. (4.17)

Proof. First of all, from (4.6), (4.7) and (4.16) we have

cM + 1 + ‖un‖ ≥ Iλ,M (un)− 1

q
〈I ′λ,M (un), un〉

=
(1

p
− 1

q

)
‖un‖p +

(λ
4
− λ

q

)
LM (un)

∫
R3

φunu
2
n dx

− aλ,M (un)

q
‖un‖p −

q − 1

q

∫
R3

h(x)un dx.

Therefore,(1

p
− 1

q

)
‖un‖p ≤ cM + 1 + ‖un‖+

(λ
q
− λ

4

)
LM (un)

∫
R3

φunu
2
n dx

+
aλ,M (un)

q
‖un‖p +

q − 1

q

∫
R3

h(x)un dx.

(4.18)

We claim that {un} is bounded. Indeed, by definition, when ‖un‖ ≥M , LM (un) =

0, χ′
(‖un‖p
Mp

)
= 0 and so (4.18) reads(1

p
− 1

q

)
‖un‖p ≤ cM + 1 + ‖un‖+

q − 1

q

∫
R3

h(x)un dx.

Thus {un} is bounded. By using (4.12), (4.13), and Hölder’s inequality,(λ
q
− λ

4

)
LM (un)

∫
R3

φunu
2
n dx ≤

∣∣λ
q
− λ

4

∣∣LM (un)A2‖un‖4 ≤ λA2M4, (4.19)

aλ,M (un)

q
‖un‖p ≤ |aλ,M (un)|‖un‖p ≤ pλA2M4−pMp = pλA2M4, (4.20)

q − 1

q

∫
R3

h(x)un dx ≤
q − 1

q

∫
R3

|h(x)un| dx

≤ |h|Ls(R3)|un|Ls′ (R3) ≤ CΛ‖un‖.
(4.21)

Let ω1 be the function taken in the proof of Lemma 4.5(ii). By (4.6), we have

Iλ,M (Mω1) ≤ Mp

p
− Mq

q
|ω1|qq.



EJDE-2023/28 SCHRÖDINGER-POISSON SYSTEMS WITH p-LAPLACIAN 13

Then there exists M1 > 0 such that Iλ,M (Mω1) < 0 for all M ≥M1. Thus

cM ≤ max
t∈[0,1]

Iλ,M (tMω1) ≤ max
t∈[0,1]

{1

p
(Mt)p − 1

q
(Mt)q|ω1|qq

}
+ max
t∈[0,1]

λ

4
(tM)4LM (tMω1)

∫
R3

φω1ω
2
1 dx

≤ C + λA2M4.

(4.22)

It follows from (4.18)–(4.22) that, for all M ≥M1,(1

p
− 1

q

)
‖un‖p ≤ C + 1 + (p+ 2)λA2M4 + (1 + CΛ)‖un‖. (4.23)

Take λ∗ = (A2M4)−1. Then it follows from (4.23) that (4.17) holds for anyM ≥M1

and λ ∈ (0, λ∗). The proof is complete. �

Proof of Theorem 4.4. Combining Lemmas 4.5–4.7 and the mountain pass theorem,
for M > 0 large enough and λ > 0 small, we can find a critical point u∗ for Iλ,M
at the level cM > 0 with ‖u∗‖ ≤ M

2 . Thus u∗ is a critical point for Iλ with
Iλ(u∗) = cM > 0. �

Acknowledgments. This research was supported by NSFC(12271373,12171326).
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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Lanxin Huang
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Email address: 812419761@qq.com

Jiabao Su

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Email address: sujb@cnu.edu.cn


	1. Introduction and statement of main results
	2. Preliminaries
	3. A solution with negative energy
	4. A solution with positive energy
	4.1. Case 6pp+2<q<p*
	4.2. Case p<q 6pp+2
	Acknowledgments

	References