Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 82, pp. 1–14.

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

DOI: 10.58997/ejde.2024.82

NORMALIZED SOLUTIONS FOR BIHARMONIC

SCHRÖDINGER EQUATIONS WITH POTENTIAL AND

GENERAL NONLINEARITY

FENGWEI ZOU, SHUAI YAO, JUNTAO SUN

Abstract. We study the existence and non-existence of normalized solutions

to the biharmonic equation

∆2u−∆u+ V (x)u+ λu = f(u) in RN .

where 0 ̸= V (x) ≤ V∞ := lim|x|→∞ V (x) ∈ (−∞,+∞] and f ∈ C(R,R) is a

nonlinearity. For the trapping case of V∞ = +∞, under some suitable assump-
tions on f , we prove that there exists a ground state as a global minimizer of

the corresponding energy functional. For the case of V∞ < +∞, under some

other assumptions on f , we prove that there exists ᾱ ≥ 0 such that a global
minimizer exists if α > ᾱ while no global minimizer exists if α < ᾱ. Moreover,

the size of ᾱ is also explored, depending on the potential V .

1. Introduction

Our starting point is the biharmonic nonlinear Schrödinger (NLS) equations

iψt − γ∆2ψ + β∆ψ + |ψ|p−2ψ = 0 in R× RN ,

ψ(x, t) = ψ0(x),
(1.1)

where ψ(x, t) : RN × [0, T ) → C is a wave function, γ, p > 0 and β ∈ R. This
equation has been introduced by Karpman and Shagalov in [11, 12] to take into
account the role of small fourth-order dispersion terms in the propagation of intense
laser beams in a bulk medium with Kerr nonlinearity, see also [6]. It has also been
used to describe the motion of a vortex filament in an incompressible fluid [7].
Equation (1.1) is Hamiltonian, and the mass and energy are conserved by the flow.

Equation (1.1) has an important class of special solutions, i.e. the standing waves.
A standing wave is a solution of the form ψ(t, x) = eiλtu(x), where λ ∈ R is a
frequency. Then the real valued function u satisfies the elliptic equation

γ∆2u− β∆u+ λu = |u|p−2u in RN . (1.2)

To study the solutions of (1.2), one can consider λ to be an unknown of the
problem. Then λ appears as a Lagrange multiplier and L2-norms of solutions are

2020 Mathematics Subject Classification. 35J20, 35J60, 35J92.

Key words and phrases. Biharmonic NLS; normalized solution; variational method.
©2024. This work is licensed under a CC BY 4.0 license.
Submitted August 2, 2024. Published December 11, 2024.

1



2 F. ZOU, S. YAO, J. SUN EJDE-2024/82

prescribed, i.e. ∫
RN

|u|2dx = α > 0,

which are usually called normalized solutions. This study seems to be particularly
meaningful from the physical point of view, since standing waves of (1.1) conserve
their mass along time. Normalized solutions to Eq. (1.2) can be identified with
critical points of the energy functional Eγ,β : H2(RN ) → R given by

Eγ,β(u) =
1

2

∫
RN

(γ|∆u|2 + β|∇u|2)dx−
∫
RN

F (u)dx,

on the set

Sγ,β(α) :=
{
u ∈ H2(RN ) :

∫
RN

u2dx = α
}
,

The study of normalized solutions to biharmonic NLS equations (1.2) has at-
tracted much attention in recent years. We refer the readers to [1, 2, 3, 5, 14, 15].
More precisely, when γ > 0, β ≤ 0 and 2 < p < p∗ := 2 + 8

N , Bonheure et al. [2]
proved the existence of a global minimizer by using the minimization method. Sub-
sequently, when γ > 0, β ≤ 0 and p∗ < p < 4∗ := 2N

N−4 , the existence of a ground
state and the multiplicity of radial solutions were obtained by the Pohozaev con-
straint method in [1]. Luo et al. [15] proved the existence of a global minimizer
when γ = 1, β ∈ R and 2 < p ≤ p∗ by using the profile decomposition of bounded
sequences in H2(RN ) established in [21]. In [3], Boussaid et al. studied (1.2) with
γ > 0, β > 0 and 2 < p ≤ p∗, which improved the results in [15] by relaxing
the extra restriction on α and β. Very recently, Luo and Yang [14] obtained the
existence of two normalized solutions for (1.2) with γ > 0, β > 0 and p∗ < p ≤ 4∗.

To the best of our knowledge, there seems to be no any results on normalized
solutions to biharmonic NLS equations with a potential and a general nonlinear-
ity in existing literature so far. Inspired by this, in this paper we focus on this
case and explore the effect of the potential on the number of normalized solutions.
Specifically, for α > 0, the problem considered in this study is as follows:

∆2u−∆u+ V (x)u+ λu = f(u) in RN ,∫
RN

u2dx = α > 0,
(1.3)

where N ≥ 5, V and f satisfy the following assumptions:

(A1) f ∈ C(R,R), f(0) = 0 and there exists ζ > 0 such that F (ζ) > 0, where
F (s) =

∫ s

0
f(t)dt for s ∈ R;

(A2) lims→0 f(s)/s = 0 and lim sup|s|→∞ |f(s)|/|s|4∗−1 <∞;

(A3) lim sup|s|→∞ f(s)s/|s|p∗ ≤ 0;

(A4) f(s)s− 2F (s) > 0 for s ̸= 0;

(A5) F (θs) ≥ θ
2(N+5)
N+1 F (s) for θ > 1 and s > 0.

(A6) 0 ̸= V (x) ≤ V∞ := lim|x|→∞ V (x) ∈ (−∞,+∞];

(A7) ε4V (εx) ≤ V (x) for ε ∈ (0, 1) and x ∈ RN .

It is clear that normalized solutions to (1.3) correspond to critical points of the
energy functional I : H →R given by

I(u) =
1

2

∫
RN

(|∆u|2 + |∇u|2 + V (x)u2)dx−
∫
RN

F (u)dx,



EJDE-2024/82 BIHARMONIC SCHRÖDINGER EQUATIONS 3

on the constraint

S(α) :=

{
u ∈ H :

∫
RN

u2dx = α

}
,

where

H :=
{
u ∈ H2(RN ) :

∫
RN

V (x)u2dx <∞
}
.

Note that H is a Hilbert space, endowed with the norm

∥u∥H :=
[ ∫

RN

(|∆u|2 + |∇u|2 + V (x)u2)dx
]1/2

.

To find solutions of (1.3), we consider the minimization problem

mα := inf
u∈S(α)

I(u).

First of all, we study the case of V∞ = +∞ and obtain the following result.

Theorem 1.1. Assume that conditions (A1)–(A3) hold and V ∈ C1(RN ) satisfies
condition (A6) with V∞ = +∞. Then mα is attained by u ∈ S(α) for any α > 0,
which is a ground state of problem (1.3).

Next, we turn to study the case of V∞ < +∞. Following from [17], we define

ᾱ = inf{α > 0 : mα < 0}, (1.4)

and need the limit

lim
s→0

F (s)

|s|2+ 8
N

< +∞. (1.5)

Then we have the following results.

Theorem 1.2. Assume that conditions (A1)–(A4) hold and V ∈ C1(RN ) satisfies
condition (A6) with V∞ < +∞. Then there exists a constant ᾱ ≥ 0 such that mα

is attained by u ∈ S(α) for α > ᾱ, which is a ground state of problem (1.3).

Theorem 1.3. Assume that conditions (A1)–(A3), (A5) hold and V satisfies con-
ditions (A6) with V∞ < +∞ and (A7). If in addition condition (1.5) holds and
ᾱ ≥ 0 is uniquely determined, then the following conclusions are true.

(i) If α > ᾱ, there exists a global minimizer with respect to mα;
(ii) If 0 < α < ᾱ, there is no global minimizer with respect to mα.

Remark 1.4. It is meaningful to point out that the condition (A4) is weaker than
the well known Ambrosetti-Rabinowitz type condition:

(AR) There exists α > 2 such that f(s)s ≥ αF (s) > 0, for all s ̸= 0.

To obtain more information, we need a stronger condition (A5). In deed, by (A5),

F (θs)

θ2s2
>
F (s)

s2
, ∀θ > 1, s ̸= 0.

This implies that s 7→ F (s)
s2 is strictly increasing for s > 0 and strictly decreasing

for s < 0. Then we have

d

ds

(F (s)
s2

)
=
f(s)s− 2F (s)

s3
> 0, for s > 0,

d

ds

(F (s)
s2

)
=
f(s)s− 2F (s)

s3
> 0, for s < 0,

which means that condition (A4) holds.



4 F. ZOU, S. YAO, J. SUN EJDE-2024/82

Moreover, we have the following result.

Theorem 1.5. Assume that conditions (A1)–(A3), (A5) hold and V satisfies con-
ditions (A6) with V∞ < +∞ and (A7). If in addition condition (1.5) holds, then
the following conclusions hold.

(i) Assume that there exists an s0 > 0 such thatf(s) ≥ 0 in [0, s0] and

inf
∥u∥2=1

∫
RN

(
|∆u|2 + |∇u|2 + V (x)u2

)
dx < 0, (1.6)

Then ᾱ = 0.
(ii) If V ∈ LN/4(RN ) satisfies ∥V ∥N/4 < S, then we have ᾱ > 0. Here, S is

defined as a Sobolev constant, i.e.,

S := inf
u∈D2,2(RN )\{0}

∥∆u∥22
∥u∥24∗

.

We wish to point out that normalized solutions of NLS equations with potential
and various types of nonlinearities has been studied by Ikoma and Miyamoto [8]
and Yang et al. [20] recently. In this article our results can been viewed as an
extension to the case of biharmonic NLS equations.

This article is structured as follows. We introduce some preliminary results in
Section 2. We give the proofs of Theorems 1.1-1.5 in Section 3.

2. Preliminary results

For sake of convenience, we set

A(u) :=

∫
RN

|∆u|2dx and B(u) :=

∫
RN

|∇u|2dx.

Then the energy functional I is rewritten as

I(u) =
1

2
A(u) +

1

2
B(u) +

1

2

∫
RN

V (x)u2dx−
∫
RN

F (u)dx.

In view of the Gagliardo-Nirenberg inequality [14], for 2 < p < 4∗, there exists a
constant CN,p > 0 such that

∥u∥pp ≤ Cp
N,p∥∆u∥

pγp

2 ∥u∥p(1−γp)
2 for u ∈ H2(RN ),

where γp := N(p−2)
4p .

Lemma 2.1. Assume that conditions (A1)–(A3) hold. Then the following state-
ments hold:

(i) For any bounded sequence {un} ⊂ H, if limn→∞ ∥un∥∞ = 0, then

lim
n→∞

∫
RN

F (un)dx = 0,

and if limn→∞ ∥un∥2+ 8
N

= 0, then

lim sup
n→∞

∫
RN

F (un)dx ≤ 0.

(ii) For any α > 0, the energy functional I is bounded from below and coercive
on S(α).



EJDE-2024/82 BIHARMONIC SCHRÖDINGER EQUATIONS 5

Proof. (i) It can be found in [9, Lemma 2.1 (i)].
(ii) According to [9, Lemma 2.1 (ii)], since V0 := infx∈RN V (x) > −∞, by using

the Poincaré inequality, there exists a constant C = C(f, α, V0) > 0 such that

I(u) =
1

2
A(u) +

1

2
B(u) +

1

2

∫
R3

V (x)u2dx−
∫
RN

F (u)dx

≥ 1

2
∥u∥2H − C(f, α, V0) for any u ∈ S(α),

which implies that the functional I is bounded from below and coercive on S(α)
for any α > 0. □

Lemma 2.2. Assume that conditions (A1)–(A3) hold. Let {un} be a bounded
sequence in H such that un ⇀ u in H. Then

lim
n→∞

∫
RN

|F (un − u) + F (u)− F (un)|dx = 0.

The proof of the above lemma is similar to [10, Lemma 3.2], so we omit it here.

Lemma 2.3 ([13, Lemma I.1]). Let {un} be a bounded sequence in H satisfying

sup
z∈RN

∫
B(z,1)

|un|2dx→ 0 as n→ ∞.

Then for q > 2, we have ∥un∥q → 0 as n→ ∞.

Lemma 2.4 ([16]). Assume that V∞ = +∞. Then the embedding H ↪→ Lq(RN ) is
compact for all 2 ≤ q < 4∗.

3. Proof of the main theorems

3.1. Case V∞ = +∞.

Proof of Theorem 1.1. According to Lemma 2.1 (ii), there exists a minimizing se-
quence {un} ⊂ S(α) of I with respect to mα such that mα = limn→∞ I(un).
Clearly, {un} is bounded in H. By Lemma 2.4, there exists ū ∈ H such that

un ⇀ ū in H,
un → ū in L2(RN ),

un → ū a.e. in RN .

This shows that ū ∈ S(α). Using Lemma 2.2 and the weak lower semi-continuity
of the H-norm, it follows that

I(ū) ≤ lim
n→∞

I(un) = mα ≤ I(ū),

which implies that mα = I(ū) and un → ū in H as n→ ∞. Therefore, ū ∈ S(α) is
a ground state solution of problem (1.3). The proof is complete. □



6 F. ZOU, S. YAO, J. SUN EJDE-2024/82

3.2. Case of V∞ < +∞. Without loss of generality, in this subsection, we may
assume that V∞ = 0 in condition (V 1). If not, we may replace (V (x), λ) by

(Ṽ (x), λ̃) := (V (x)− V∞, λ+ V∞),

and problem (1.3) becomes the equivalent problem

∆2u−∆u+ Ṽ (x)u+ λ̃u = f(u), in RN ,∫
RN

u2dx = α.

Lemma 3.1. Assume that conditions (A1)–(A3), (A6) with V∞ = 0 hold. Then
we have

(i) mα ≤ m∞
α ≤ 0 for any α ≥ 0,

(ii) mα ≤ mβ +mα−β for α > β > 0,
(iii) m∞

α and mα are non-increasing on α ≥ 0,
(iv) α 7→ mα is continuous for α > 0.

The proof of the above lemma is almost the same as [8, Lemma 2.5], se we omit
it here.

Lemma 3.2. Assume that conditions (A1)–(A3), (A6) hold. Then there exists
α∗ > 0 such that mα < 0 for all α > α∗. Assume that in addition condition (1.5)
holds, for α > 0 small enough, we have mα = 0.

Proof. From condition (A1), there exists u ∈ H2(RN ) such that
∫
RN F (u)dx > 0.

For any α > 0, set uα := u(α−1/N · ∥u∥2/N2 · x) ∈ Sα. Since

I∞(uα)

=
1

2

∫
RN

|∆uα|2dx+
1

2

∫
RN

|∇uα|2dx−
∫
RN

F (uα)dx

=
α

N−4
N

2∥u∥2(N−4)/N
2

∫
RN

|∆u|2dx+
α

N−2
N

2∥u∥2(N−2)/N
2

∫
RN

|∇u|2dx− α

∥u∥22

∫
RN

F (u)dx

=: C1α
N−4
N + C2α

N−2
N − C3α =: g(α),

it follows that m∞
α ≤ I∞(uα) = g(α) < 0 for sufficiently large α > 0.

By condition (1.5), there exists Cf > 0 such that F (s) ≤ Cf |s|2+8/N for any
s ∈ R. By the Gagliardo-Nirenberg inequality,∫

RN

F (u)dx ≤ CfCp∗

N,p∗α
8/N∥∆u∥22 for all u ∈ Sα.

For any α > 0 small enough such that CfCp∗

N,p∗α8/N ≤ 1/4, we have

I(u) ≥ 1

4
∥∆u∥22 > 0,

and thus mα ≥ 0. It follows that mα = 0 for α > 0 small enough. The proof is
complete. □

For u ∈ S(1), we set

ut(x) := tN/2u(tx).



EJDE-2024/82 BIHARMONIC SCHRÖDINGER EQUATIONS 7

It is clear that ut ∈ S(1) and
√
αut ∈ S(α). Define the fibering map gα,u(t) :

(0,+∞) → R by

gα,u(t) := I(
√
αut)

=
αt2

2
A(u) +

α2t

2
B(u) +

α

2

∫
RN

V (x/t)|u|2dx− 1

tN

∫
RN

F
(√

αtNu
)
dx.

By calculating the first derivative of gα,u, we have

g′α,u(t) = tαA(u) + αB(u)− α

2t2

∫
RN

⟨∇V (x/t), x⟩|u|2dx

− N

2tN+1

∫
RN

[
f
(√

αtNu
)√

αtNu− 2F
(√

αtNu
)]
dx,

and

g′α,u(1) = αA(u) + αB(u)− α

2

∫
RN

⟨∇V (x), x⟩u2dx

− N

2

∫
RN

[f(
√
αu)

√
αu− 2F (

√
αu)]dx

=: Qα(u).

Then we have the Pohozaev identity.

Lemma 3.3. Let u ∈ S(1) be a critical point of the functional I restricted to S(1).
Then

2A(u) +B(u)− 1

2

∫
RN

⟨∇V (x), x⟩u2dx− N

2

∫
RN

[f(u)u− 2F (u)]dx = 0. (3.1)

Proof. Since u ∈ S(1) is a critical point of I restricted to S(1), there exists a
Lagrange multiplier λ ∈ R such that

∆2u−∆u+ V (x)u+ λu = f(u). (3.2)

Multiplying (3.2) by u and integrating, we obtain∫
RN

(|∆u|2 + |∇u|2 + (V (x) + λ)u2)dx =

∫
RN

f(u)udx.

From [19, Lemma 2.2], we have

N − 4

2

∫
RN

|∆u|2dx+
N − 2

2

∫
RN

|∇u|2dx+
N

2

∫
RN

(V (x) +N⟨∇V (x), x⟩+ λ)u2dx

= N

∫
RN

F (u)dx.

Therefore, combining these two equalities above, we obtain (3.1). The proof is
complete. □

Now, we define the set

Mα := {u ∈ S(1) : Qα(u) = 0},
and the functional h(α) : [0,+∞) → R by

h(α) :=
α

2
A(u) +

α

2
B(u) +

α

2

∫
RN

V (x)u2dx−
∫
RN

F (
√
αu)dx

for u ∈ Mα. Then we have the following lemmas.



8 F. ZOU, S. YAO, J. SUN EJDE-2024/82

Lemma 3.4. For each α > 0, it holds

m̃α := inf
u∈Mα

I(
√
αu) = mα = inf

u∈S(α)
I(u).

Proof. According to the definition of m̃α, obviously mα ≤ m̃α. In addition, for
any u ∈ S(α) with Qα(

1√
α
u) = 0, we have m̃α ≤ I(

√
α 1√

α
u) = I(u). Taking the

infimum, we obtain m̃α ≤ mα. Therefore, m̃α = mα. The proof is complete. □

Lemma 3.5. Assume that conditions (A1)–(A4) hold. In addition, let V ∈ C1(RN )
satisfy condition (A6). Then the following conclusions are true:

(i) the function α 7→ h(α)
α is decreasing for all α > 0,

(ii) if mα is achieved for α > 0, then the function α 7→ mα

α is decreasing for all
α > 0.

Proof. (i) For u ∈ Mα fixed, define

J(α) :=
h(α)

α
for α > 0.

Then J(α) ∈ C1(R) and J ′(α) = h′(α)α−h(α)
α2 . By calculating the first derivative of

h(α) one has

h′(α) =
1

2
A(u) +

1

2
B(u) +

1

2

∫
RN

V (x)|u|2dx− 1

2
√
α

∫
RN

f(
√
αu)udx,

which implies that

h′(α)α− h(α) = −1

2

∫
RN

[
f(
√
αu)

√
αu− 2F (

√
αu)

]
dx. (3.3)

By condition (A4), we obtain

f(
√
αu)

√
αu− 2F (

√
αu) > 0. (3.4)

Thus, it follows from (3.3) and (3.4) that h′(α)α− h(α) < 0, so (i) is valid.
(ii) Fix 0 < α1 < α2, and let ui ∈ Mαi satisfy mαi = I(

√
αiui) for i = 1, 2.

Then from (i) and the definition of mα, it follows that

mα2

α2
≤
I(
√
α2u2)

α2
=
h(α2)

α2
<
h(α1)

α1
=
I(
√
α1u1)

α1
=
mα1

α1
.

This indicates that the function α 7→ mα

α is decreasing for all α > 0. The proof is
complete. □

As a direct consequence of Lemma 3.5, we have the following lemma.

Lemma 3.6. Assume that conditions (A1)–(A4) hold. In addition, let V ∈ C1(RN )
satisfy condition (A6). If mα is attained for some α > 0, then for any α1, α2 ∈
(ᾱ,+∞), we have

mα2
< mα1

+mα2−α1
.

Lemma 3.7. Assume that ((A1)–(A4) hold. In addition, let V ∈ C1(RN ) satisfy
condition (A6). Let {un} ⊂ S(α) be a minimizing sequence of I with respect to
m(α) for α > ᾱ, then one of the following conclusions hold:

(i)

lim sup
n→∞

sup
z∈RN

∫
B(z,1)

|un|2dx = 0;



EJDE-2024/82 BIHARMONIC SCHRÖDINGER EQUATIONS 9

(ii) Taking a sequence if necessary, there exist u ∈ S(α) and a family {yn} ⊂
RN such that

un(· − yn) → u in H as n→ ∞.

Proof. Assume that (i) does not hold. Then

0 < lim sup
n→∞

sup
z∈RN

∫
B(z,1)

|un|2dx ≤ α <∞.

Taking a subsequence if necessary, there exists a family {yn} ⊂ RN such that

0 < lim
n→∞

∫
B(0,1)

|un(x− yn)|2dx <∞.

By Lemma 2.1, {un} is a bounded sequence in H. Thus, up to a subsequence, there
exists u ∈ H such that

un(· − yn)⇀ u in H,
un(· − yn) → u in L2

loc(RN ),

un(· − yn) → u a.e. in RN ,

which implies that 0 < ∥u∥22 ≤ α. Set η := ∥u∥22 and vn := un(· − yn) − u. It is
clear that vn ⇀ 0 in H as n → ∞. Moreover, it follows from Brezis-Lieb theorem
[4] and Lemma 2.2 that

I(un) = I(u) + I(vn) + o(1).

Next, we prove that η = α. Otherwise, if 0 < η < α, using Brezis-Lieb theorem [4]
again, we have

∥un∥22 = ∥u+ vn∥22 = ∥u∥22 + ∥vn∥22 + o(1), (3.5)

which implies that ∥vn∥22 = α−η+o(1) > 0. To obtain a contradiction, we consider
two separate cases. If mη is not attained by u, then by Lemma 3.1, we have

mα = I(un) + o(1)

= I(u) + I(vn) + o(1)

> mη + I(vn) + o(1)

≥ mη +mα−η ≥ mα,

which is a contradiction. If mη is attained by u, then from Lemma 3.6, it follows
that

mα = I(u) + I(vn) + o(1)

= mη + I(vn) + o(1)

≥ mη +mα−η > mα,

which is also a contradiction. Hence, η = ∥u∥22 = α and u ∈ S(α).
Note that

I(un) = I(u) + I(vn) + o(1) ≥ mα + I(vn) + o(1),

which means that

lim
n→∞

I(vn) ≤ 0. (3.6)



10 F. ZOU, S. YAO, J. SUN EJDE-2024/82

By (3.5), ∥vn∥22 → 0 as n→ ∞. Then for any ε > 0, there exists R > 0 such that

|
∫
RN

V (x)v2ndx| ≤
∫
BR(0)

V (x)v2ndx+

∫
RN\BR(0)

V (x)v2ndx

≤ sup
BR(0)

|V (x)|
∫
BR

v2ndx+ sup
RN\BR(0)

|V (x)|
∫
RN

v2ndx

≤ Cε,

where C > 0 is a constant. This indicates that

lim
n→∞

∫
RN

V (x)v2ndx = 0. (3.7)

Applying Lemmas 2.1 and 2.3 leads to

lim
n→∞

∫
RN

F (vn)dx ≤ 0. (3.8)

So, by (3.6)–(3.8), we have

1

2
lim
n→∞

∫
RN

(|∆vn|2 + |∇vn|2)dx

≤ lim
n→∞

I(vn) + lim
n→∞

(∫
RN

F (vn)dx−
∫
RN

V (x)v2ndx
)
≤ 0,

which implies that vn → 0 in H. Therefore, limn→∞ un(· − yn) = u in H. The
proof is complete. □

Proof of Theorem 1.2. By Lemma 2.1, let {un} ⊂ S(α) be a bounded minimizing
sequence of I with respect to mα. It is sufficient to show that {un} satisfies Lemma
3.7 (ii). Otherwise,

lim sup
n→∞

sup
z∈RN

∫
B(z,1)

u2ndx = 0. (3.9)

By (3.9) and Lemma 2.3, we have

un → 0 in Lp(RN ) for 2 < p < 4∗. (3.10)

Then it follows from Lemma 2.1 that

lim
n→∞

∫
RN

F (un)dx ≤ 0. (3.11)

In addition, since lim|x|→∞ V (x) = 0, for each ε > 0, there exists a M > 0 such
that

|V (x)| < ε for |x| > M. (3.12)

According to (3.10), we know that

un → 0 in L2(BM (0)). (3.13)

Then combining (3.12) and (3.13), we deduce that∣∣ ∫
RN

V (x)u2ndx
∣∣ ≤ sup

BM (0)

|V (x)|
∫
BM (0)

u2ndx+ sup
RN\BM (0)

|V (x)|
∫
RN

u2ndx ≤ ε,

which means that

lim
n→∞

∫
RN

V (x)u2ndx = 0. (3.14)



EJDE-2024/82 BIHARMONIC SCHRÖDINGER EQUATIONS 11

Hence, by (3.11) and (3.14) one has

mα = lim
n→∞

I(un) ≥ lim
n→∞

(1
2

∫
RN

V (x)u2ndx−
∫
RN

F (un)dx
)
≥ 0,

contradicting to mα < 0. Therefore, there exists a global minimizer u such that
I(u) = mα, that is, u ∈ S(α) is a ground state solution of problem (1.3). The proof
is complete. □

Lemma 3.8. Assume that conditions (A1)–(A3), (A5)–(A7) and (1.5) hold. Then
for any α > ᾱ, we have

(i) mlα ≤ lmα for any l > 1,
(ii) if mα is attained, then mlα < lmα for all l > 1.

Proof. (i) For each ε > 0, there exists u ∈ S(α) such that I(u) < mα + ε. Set

ul := l
N+1

2 u(lx), we have ∫
RN

u2l dx = l

∫
RN

u2dx = lα,

which implies that ul ∈ S(lα). Then it follows from (A5) and (A7) that for all
l > 1,

mlα ≤ I(ul)

=
l5

2
∥∆u∥22 +

l3

2
∥∇u∥22 +

l

2

∫
RN

V (x/l)u2dx− 1

lN

∫
RN

F (l
N+1

2 u)dx

= l5
(1
2
∥∆u∥22 +

1

2l2
∥∇u∥22 +

1

2l4

∫
RN

V (x/l)u2dx− 1

lN+5

∫
RN

F (l
N+1

2 u)dx
)

= l5
[
I(u) +

1

2
(
1

l2
− 1)∥∇u∥22 +

1

2

∫
RN

( 1

l4
V (x/l)− V (x)

)
u2dx

+

∫
RN

(
F (u)− 1

lN+5
F (l

N+1
2 u)

)
dx

]
< l5I(u) < l5(mα + ε).

This implies that mlα ≤ l5mα ≤ lmα for all l > 1, since ε > 0 is arbitrary.
(ii) Let mα be attained by some u ∈ S(α), i.e. I(u) = mα. According to (i), we

have
mlα < lmα for any l > 1.

The proof is complete. □

As an immediate consequence of Lemma 3.8, we have the following lemma.

Lemma 3.9. Assume that conditions (A1)–(A3), (A5)–(A7) and (1.5) hold. If mα

is attained for α > ᾱ, then for each α1, α2 ∈ (ᾱ,∞),

mα2 < mα1 +mα2−α1 .

Proof of Theorem 1.3. (i) Suppose by contradiction that there exists a global min-
imizer of the energy functional I with respect to mα for 0 < α < ᾱ. According to
( 1.4), mα = 0 when 0 < α < ᾱ. Then we infer from Lemma 3.8 (ii) that

0 = mα > mᾱ,

which contradicts with Lemma 3.1 (iii)-(iv). Hence, mα is not attained for 0 < α <
ᾱ.



12 F. ZOU, S. YAO, J. SUN EJDE-2024/82

(ii) By Lemma 3.2, we have mα < 0 for α > ᾱ. It follows from Lemma 2.1 that
there exists a minimizing sequence {un} ⊂ S(α) such that limn→∞ I(un) = mα.
Next, applying the argument of Theorem 1.2, by Lemmas 3.1, 2.3 and 3.9, there
exists a global minimizer u such that I(u) = mα. The proof is complete. □

Proof of Theorem 1.5. (i) By (1.6), there exists a u ∈ C∞
0 (RN ) such that ∥u∥22 = 1

and ∫
RN

(
|∆u|2 + |∇u|2 + V (x)u2

)
dx < 0.

Replacing with |u| and thus we can assume that u is non-negative. Let α ∈
(0, s20/∥u∥2∞). Clearly,

√
αu ∈ S(α) and F (

√
αu) ≥ 0. Then there exists α0 ∈

(0, s20/∥u∥2∞) such that for α < α0,

mα ≤ I(
√
αu) ≤ α

2

∫
RN

(|∆u|2 + |∇u|2 + V (x)u2)dx < 0.

By the monotonicity of mα in Lemma 3.1, we obtain that mα < 0 for all α > 0, so
ᾱ = 0.

(ii) By condition (1.5), there exists Cf > 0 such that F (t) ≤ Cf |t|2+
8
N for any

t ∈ R. According to the Gagliardo-Nirenberg inequality, we have∫
RN

F (u)dx ≤ CfC
2+ 8

N

N,2+ 8
N

α4/N∥∆u∥22 for all u ∈ S(α). (3.15)

By [18, Lemma 2.2], we recall that∣∣ ∫
RN

V (x)u2dx
∣∣ ≤ ∥V ∥N/4∥u∥24∗ ≤ U−1∥V ∥N/4∥∆u∥22, (3.16)

where S is defined as a Sobolev constant, namely,

S := inf
u∈D2,2(RN )\{0}

∥∆u∥22
∥u∥24∗

. (3.17)

Define

α1 :=
(1− S−1∥V ∥N/4

2CfC
2+ 8

N

N,2+ 8
N

)N/4

.

It is clear that α1 > 0 when ∥V ∥N/4 < S. Then for u ∈ S(α) with α ∈ (0, α1), it
follows from (3.15) and (3.16) that

I(u) =
1

2
A(u) +

1

2
B(u) +

1

2

∫
RN

V (x)u2dx−
∫
RN

F (u)dx

≥ 1

2
B(u) +

1

2

(
1− S−1∥V ∥N

4
− 2CfC

2+ 8
N

N,2+ 8
N

α
4
N

)
A(u) ≥ 0,

which indicates that mα = 0 and ᾱ ≥ α > 0 from the monotonicity of mα. The
proof is complete. □

Acknowledgments. J. Sun was supported by the National Natural Science Foun-
dation of China (Grant No. 12371174) and by the Shandong Provincial Natural
Science Foundation (Grant No. ZR2020JQ01).



EJDE-2024/82 BIHARMONIC SCHRÖDINGER EQUATIONS 13

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Fengwei Zou
School of Mathematics and Statistics, Shandong University of Technology, Shandong,

Zibo 255049, China
Email address: zfw746265367@163.com



14 F. ZOU, S. YAO, J. SUN EJDE-2024/82

Shuai Yao

School of Mathematics and Statistics, Shandong University of Technology, Shandong,

Zibo 255049, China
Email address: shyao@sdut.edu.cn

Juntao Sun (corresponding author)
School of Mathematics and Statistics, Shandong University of Technology, Shandong,

Zibo 255049, China

Email address: jtsun@sdut.edu.cn


	1. Introduction
	2. Preliminary results
	3. Proof of the main theorems
	3.1. Case V=+
	3.2. Case of V<+
	Acknowledgments

	References