Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 49, pp. 1–11.

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

DOI: 10.58997/ejde.2024.49

EXISTENCE OF NON-GLOBAL SOLUTIONS TO BI-HARMONIC

INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATIONS

WITHOUT GAUGE INVARIANCE

TAREK SAANOUNI

Abstract. This work shows the existence of non-global mass and energy solu-
tions to inhomogeneous nonlinear bi-harmonic Schrödinger problems without

gauge invariance.

1. Introduction

This note studies the initial valued problem for the inhomogeneous non-linear
fourth-order Schrödinger equation

iut +∆2u = λ|x|−τ |u|p,
u(0, ·) = νu0,

(1.1)

where u : (t, x) ∈ R × RN → C for some integer N ≥ 1 is the wave function. The
source term satisfies 0 ̸= λ ∈ C, p > 1, and τ > 0. The datum satisfies ν ∈ R.

The fourth-order Schrödinger problem takes into account the role of small fourth-
order dispersion terms in the propagation of intense laser beams in a bulk medium
with a Kerr non-linearity [13, 14]. It was also considered in [12, 22, 1] to study
the stability of solitons in magnetic materials once the effective quasi particle mass
becomes infinite.

The inhomogeneous nonlinear Schrödinger equation (1.1) is called non gauge
invariant because the source term N (u) := |x|−τ |u|p satisfies

N (eiθu) ̸= eiθN (u), θ ∈ R− 2πZ. (1.2)

Well-posedness issues of the inhomogeneous non-linear Schrödinger equation
with a gauge invariant source term

iut +∆2u = ±|x|−τ |u|p−1u, (1.3)

were investigated by many authors in the previous few years. Indeed, a local theory
was developed in [10] using contraction mapping argument via Strichartz inequal-
ities and revisited in [15]. A small data global existence result was proved in [11].
A sharp dichotomy of global versus non-global existence of solutions by using the
existence of ground states is given by the author in [19]. The energy scattering in

2020 Mathematics Subject Classification. 35Q55, 35A01.
Key words and phrases. Inhomogeneous fourth-order Schrödinger problem;

nonlinear equations; non-global solutions.
©2024. This work is licensed under a CC BY 4.0 license.
Submitted July 24, 2024. Published September 3, 2024.

1



2 T. SAANOUNI EJDE-2024/49

the inter-critical regime was first proved by the author [17, 18] for radial data in
space dimensions N ≥ 5 and revisited in [4, 7] for non-radial data and low space
dimensions. The existence of non-global solutions was investigated in [5, 6] for the
mass-critical regime with negative energy and by the author in [3] for the inter-
critical regime. In the energy critical regime, a local theory was developed recently
by the author [21, 20], see also [2]. To the author knowledge, there is no work
dealing with the inhomogeneous bi-harmonic Shrödinger equation with non-gauge
invariance (1.3).

This note aims to present some differences between the gauge invariant problem
(1.1) and the non-gauge invariant one (1.3). Indeed, we prove the existence of non
global mass-solutions for 1 < p < 1 + 4−τ

N and the existence non global energy-

solutions for 1 < p < 1 + 2(4−τ)
N−4 with small data. These results are known to be

false for the gauge invariant problem (1.3). To the author knowledge, this work
is the first one dealing with the inhomogeneous nonlinear bi-harmonic Schrödinger
equation with non-gauge invariant source term (1.1).

The plan of this note is as follows. Section 2 contains the main results and some
useful inequalities. Sections 3-4 presents the proof of the main results.

We denote the standard Lebesgue and Sobolev spaces and norms by

Lr := Lr(RN ), H2 := {f ∈ L2, ∆f ∈ L2},

∥ · ∥r := ∥ · ∥Lr , ∥ · ∥ := ∥ · ∥2, ∥ · ∥H2 :=
(
∥ · ∥2 + ∥∆ · ∥2

)1/2

.

We define the ball of RN with center at the origin and radius R > 0 by B(R) :=
{x ∈ RN , |x| < R}, and its complement by Bc(R) := {x ∈ RN , x /∈ B(R)}. The
annulus of RN with radii 0 < R′ < R is C(R′, R) := {x ∈ RN , R′ < |x| < R}. Also
x− is a real number close to x such that x > x− and r′ := r/(r − 1) is the Hölder
conjugate of r > 1.

2. Background and main result

This section contains the main contribution of this note and some useful standard
estimates.

2.1. Preliminaries. Let us denote the free bi-harmonic Schrödinger kernel by

eit∆
2

u := F−1
(
eit|·|

4

Fu
)
. (2.1)

where F is the Fourrier transform. Thanks to the Duhamel formula, solutions to
(1.1) are fix points of the integral operator

f(u(t)) := eit∆
2

u0 − iλ

∫ t

0

ei(t−s)∆2(
|x|−τ |u|p

)
ds. (2.2)

If u resolves (1.1), then so does the family uκ := κ
4−τ
p−1 u(κ4·, κ·), κ > 0. Moreover,

there is only one invariant Sobolev norm under the above dilatation, precisely

∥uκ(t)∥Ḣsc = ∥u(κ4t)∥Ḣsc , sc :=
N

2
− 4− τ

p− 1
.

In contrast to (1.3), the mass and energy are not conserved quantities. The above
problem (1.1) is said to be

mass-sub-critical if sc < 0 ⇔ p < pc := 1 +
2(4− τ)

N
; (2.3)



EJDE-2024/49 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 3

energy-sub-critical if sc < 2 ⇔ p < pc := 1 +
2(4− τ)

N − 4
. (2.4)

In (2.4), we take pc = ∞ if 1 ≤ N ≤ 4. For future convenience, we recall the
so-called Strichartz inequalities.

Definition 2.1. A couple of real numbers (q, r) is said to be admissible if

2 ≤ r <
2N

N − 4
, 2 ≤ q, r ≤ ∞ and N

(1
2
− 1

r

)
=

4

q
.

Denote the set of admissible pairs by Λ. If I is a time slab, one denotes the
Strichartz spaces as

Ω(I) := ∩(q,r)∈ΛL
q(I, Lr).

Now we recall some Strichartz inequalities [16, 9].

Proposition 2.2. Let N ≥ 1 and T > 0. Then,

(1) sup(q,r)∈Λ ∥u∥Lq
T (Lr) ≲ ∥u0∥+ inf(q̃,r̃)∈Λ ∥iut +∆2u∥

Lq̃′
T (Lr̃′ )

;

(2) sup(q,r)∈Λ ∥∆u∥Lq
T (Lr) ≲ ∥∆u0∥+ ∥iut +∆2u∥

L2
T (Ẇ

1, 2N
2+N )

for all N ≥ 3;

(3) sup(q,r)∈Λ ∥u∥Lq
T (Lr) ≲ ∥u0∥Ḣs + inf(q̃,r̃)∈Λ ∥iut +∆2u∥

Lq̃′
T (Lr̃′ )

.

The existence of L2 and energy solutions to (1.1) follows as in [10].

Proposition 2.3. Let N ≥ 1.

(1) If 0 < τ < min{N, 4}, 1 < p < pc and u0 ∈ L2, then there exist T :=
TN,τ,p,∥u0∥ > 0 and a unique local solution of (1.1), in the space

C([0, T ], L2) ∩ Ω(0, T ).

(2) If N ≥ 3, 0 < τ < min{4, N2 }, max{1, 2(1−τ)
N } < p < pc and u0 ∈ H2,

there exist T := TN,τ,p,∥u0∥H2
> 0 and a unique local solution of (1.1), in

the space
C([0, T ], H2) ∩(q,r)∈Λ L

q
T (W

2,r).

We recall the so-called weak solution to the Schrödinger problem (1.1).

Definition 2.4. For T > 0, a function u ∈ L1
loc([0, T ]×RN ) is said a weak solution

of (1.1) on [0, T ) if for each v ∈ C∞
0 ([0, T )× RN ),∫ T

0

∫
RN

u
(
− i∂tv +∆2v

)
dx dt = iν

∫
RN

u0v(0, ·) dx+ λ

∫ T

0

∫
RN

|x|−τ |u|pv dx dt.

(2.5)

From now on, we hide the time variable t for simplicity, showing it only when
necessary.

2.2. Main results. The first contribution of this note is the non-global well-
posedness of (1.1) in L2.

Theorem 2.5. Let N ≥ 1, 0 < τ < min{N, 4} and 1 < p ≤ 1 + 4−τ
N . Then, there

is u0 ∈ L2 such that the unique maximal solution u ∈ C([0, T+), L2) of (1.1) with
ν = 1 is non-global.

In view of the results stated in the above theorem, some comments are in order.

• The existence of the local solution is given by Proposition 2.3.



4 T. SAANOUNI EJDE-2024/49

• In the case of a gauge invariant inhomogeneous source term, namely (1.3),
any mass-sub-critical solution is global. So, the above result makes an
essential difference between (1.1) and (1.3).

• The choice ν = 1 is not necessary, indeed, in the proof we can pick ν = −1
with some change in the choice of the datum.

• With standard method, Theorem 2.5 implies that limT+ ∥u(t)∥ = ∞.
• For the heat equation, the Fujita exponent p = 1+ 4−τ

N gives the threshold
between the small data global existence and blow-up of solutions [8].

The second contribution of this note is the non-global well-posedness of (1.1) in
H2.

Theorem 2.6. Let N ≥ 3 and τ < min{4, N2 }. Take λ = 1 and max{1, 2(1−τ)
N } <

p < pc. There exist ν > 0 and u0 ∈ H2 such that the unique maximal solution
u ∈ C([0, T+), H2) of (1.1) is non global.

In view of the above theorem, some comments are in order.

• The existence of the local solution is given by Proposition 2.3.
• in the proof, we see that Theorem 2.6 holds for any ν > ν0 > 0. This gives
a blow-up result for small datum. This makes a difference with (1.3), where
the global existence is known for small data.

• With standard method, Theorem 2.6 implies that limT+ ∥∆u(t)∥ = ∞.

2.3. Sketch of the proofs. The proof of Theorem 2.5 is based on the two next
results.

Proposition 2.7. The solution given by Proposition 2.3 is a weak solution to (1.1).

Proposition 2.8. Letting 1 < p < 1 + 4−τ
N , there is a certain 0 ̸= u0 ∈ L2 such

that if u is a global weak solution to (1.1), and then u = 0.

Indeed, letting u ∈ C([0, T+), L2) be a maximal solution to (1.1). If T+ = ∞,
then by Proposition 2.7, u is a global weak solution to (1.1). So, by Proposition
2.8, u = 0, which contradicts u0 ̸= 0. This completes the proof of Theorem 2.5.
The proof of Theorem 2.6 is essentially based on Proposition 2.7 and Lemma 4.1.

3. No global mass solutions

In this section, we fix ν = 1 and we prove Theorem 2.5. It is sufficient to establish
Propositions 2.7 and 2.8.

3.1. Weak solutions. In this sub-section, we prove Proposition 2.7. Let v ∈
C∞

0 ([0, T )×RN ), for some T > 0 and u ∈ C([0, T ], L2(RN ))∩Ω(0, T ) be a solution
to (2.2). So, u ∈ L1

loc([0, T ]× RN ). By a density argument, we have∫ T

0

∫
RN

eit∆
2

u0(−i∂tv +∆2v) dx dt = i

∫
RN

u0(x)v(0, x) dx. (3.1)

So, by (2.2), (2.5) and (3.1), it is sufficient to prove that∫ T

0

∫
RN

w(−i∂tv +∆2v) dx dt = λ

∫ T

0

∫
RN

|x|−τ |u|pv dx dt; (3.2)

w := −iλ

∫ t

0

ei(t−s)∆2(
|x|−τ |u|p

)
ds. (3.3)



EJDE-2024/49 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 5

With a density argument, we take

un ∈ C∞
0 ([0, T )× RN ), lim

n
∥un − u∥Ω(0,T ) = 0. (3.4)

Let also define the sequence

wn := −iλ

∫ t

0

ei(t−s)∆2(
|x|−τ |un|p

)
ds. (3.5)

Now, by Strichartz and Hölder inequalities and taking account of [10, Lemma
3.1], for some α1, α2 > 0, we write

∥wn − w∥L∞([0,T ),L2)

≲ ∥|x|−τ
(
|un|p − |u|p

)
∥Ω′(0,T )

≲ ∥|x|−τ
(
|un|p−1 + |u|p−1

)
(un − u)∥Ω′(0,T )

≲
(
Tα1 + Tα2

)(
∥un∥p−1

Ω(0,T ) + ∥u∥p−1
Ω(0,T )

)
∥un − u∥Ω(0,T )

→ 0, as n→ ∞.

(3.6)

Since wn(0, ·) = 0, the first term of the left-hand side of (3.2) reads

−i

∫ T

0

∫
RN

w∂tv dx dt = −i lim
n

∫ T

0

∫
RN

wn∂tv dx dt

= i lim
n

∫ T

0

∫
RN

∂twnv dx dt.

(3.7)

Moreover, by Strichartz and Hölder inequalities and taking account of [10, Lemma
3.2], we write for some β1, β2 > 0,

∥∆wn∥L∞([0,T ),L2) ≲ ∥|x|−τ |un|p−1|∇un|+ |x|−τ−1|un|p∥
L1

T (L
2N

2+N )

≲
(
T β1 + T β2

)(
∥∆un∥Ω(0,T ) + ∥un∥Ω(0,T )

)p
.

(3.8)

So, wn ∈ C([0, T ), H2) and we have the equality in C([0, T ), H−2),

i∂twn = −∆2wn + λ|x|−τ |un|p. (3.9)

Moreover, since 0 < τ < N , by Hölder inequality for supp(un(0, ·)) ⊂ B(Rn),

∥∂twn(t)∥ =
∥∥∂t ∫ t

0

eis∆
2(
|x|−τ |un(t− s)|p

)
ds
∥∥

≤ ∥eit∆
2(
|x|−τ |un(0)|p

)
∥+ ∥

∫ t

0

eis∆
2(
|x|−τ∂t|un|p(t− s)

)
ds∥

≲ ∥|x|−τupn(0)∥+ ∥|x|−τ∂t|un|p∥Ω′(0,t)

≲ ∥|x|−τ∥
L(N

τ
)− (B(Rn))

∥un(0)∥p∞R
N
r′
n

+
(
Tα1 + Tα2

)
∥un∥p−1

Ω(0,T )∥∂tun∥Ω(0,t)

≲ Cn +
(
Tα1 + Tα2

)
∥un∥p−1

Ω(0,T )∥∂tun∥Ω(0,t).

(3.10)



6 T. SAANOUNI EJDE-2024/49

Thus, un ∈ C∞
0 ([0, T ) × RN ) implies that wn ∈ C1([0, T ), L2) and (3.9) gives

wn ∈ C([0, T ), H2). So, (3.7) and (3.9) imply that

−i

∫ T

0

∫
RN

w∂tv dx dt = lim
n

∫ T

0

∫
RN

(
−∆2wn + λ|x|−τ |un|p

)
v dx dt

= − lim
n

∫ T

0

∫
RN

wn∆
2v dx dt

+ λ lim
n

∫ T

0

∫
RN

|x|−τ |un|pv dx dt.

(3.11)

Furthermore, by Hölder inequality via (3.6), we have∣∣ ∫ T

0

∫
RN

(wn − w)∆2v dx dt
∣∣ ≤ T∥wn − w∥L∞

T (L2)∥∆2v∥L∞
T (L2) → 0. (3.12)

Also, by Hölder and Strichartz inequalities and arguing as previously, we have∣∣ ∫ T

0

|x|−τ
(
|un|p − |u|p

)
v dx dt

∣∣
≤ ∥|x|−τ

(
|un|p − |u|p

)
∥Ω′(0,T )∥v∥Ω(0,T )

≲ ∥|x|−τ
(
|un|p − |u|p

)
∥Ω′(0,T )∥v∥Ω(0,T )

≲
(
Tα1 + Tα2

)(
∥un∥p−1

Ω(0,T ) + ∥u∥p−1
Ω(0,T )

)
∥un − u∥Ω(0,T )∥v∥Ω(0,T ) → 0.

(3.13)

So, by (3.11), (3.12) and (3.13), we obtain

−i

∫ T

0

∫
RN

w∂tv dx dt = −
∫ T

0

∫
RN

w∆2v dx dt+λ

∫ T

0

∫
RN

|x|−τ |u|pv dx dt. (3.14)

Then by (3.2) and (3.14) the proof is complete.

3.2. Vanishing solutions. This sub-section proves Proposition 2.8. Let us define
some smooth functions:

η ∈ C∞
0 ([0,∞)), 0 ≤ η ≤ 1, η :≡

{
1, on [0, 12 ];

0, on [1,∞).
(3.15)

Also we define ϕ ∈ C∞
0 ([0,∞)) such that

0 ≤ ϕ ≤ 1, sup
{x∈B(1)}

|∇ϕ(x)|2

ϕ(x)
≲ 1, ϕ :≡

{
1, on B( 12 );

0, on Bc(1).
(3.16)

For R > 0, we define the cut-off functions

ηR :≡ η
( ·
R4

)
, ϕR :≡ ϕ

( ·
R

)
, ψR :≡ ηRϕR. (3.17)

Assume that

ℜ(u0) = 0, u0 ∈ L1 and ℜ(λ)
∫
RN

ℑ(u0) dx < 0. (3.18)

Without loss of generality, we assume that

ℜ(λ) > 0 and

∫
RN

ℑ(u0) dx < 0. (3.19)



EJDE-2024/49 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 7

For q > max{1, 3− 3
p}, let us denote

JR := JR(p, q) := ℜ
(
λ

∫ R4

0

∫
B(R)

|x|−τ |u|pψq
R dx dt

)
. (3.20)

Since u is a global weak solution to (1.1), by (2.5), we have

∫ R4

0

∫
RN

u
(
− i∂t(ψ

q
R) + ∆2(ψq

R)
)
dx dt

= i

∫
RN

u0ψ
q
R(0, ·) dx+ λ

∫ R4

0

∫
RN

|x|−τ |u|pψq
R dx dt.

(3.21)

So, (3.21) via (3.17), gives

JR = ℑ
( ∫

B(R)

u0ψ
q
R(0, ·) dx

)
+ ℑ

( ∫ R4

0

∫
B(R)

u∂t(ψ
q
R) dx dt

)
+ ℜ

(∫ R4

0

∫
B(R)

u∆2(ψq
R) dx dt

)
=

∫
B(R)

ℑu0ψq
R(0, ·) dx+

∫ R4

0

∫
B(R)

ℑu∂t(ψq
R) dx dt

+

∫ R4

0

∫
B(R)

ℜu∆2(ψq
R) dx dt.

(3.22)

Now, by (3.19), for R >> 1, we have

JR <

∫ R4

0

∫
B(R)

ℑu∂t(ψq
R) dx dt+

∫ R4

0

∫
B(R)

ℜu∆2(ψq
R) dx dt

≲
∫ R4

0

∫
B(R)

|u|ψq−1
R |∂tψR| dx dt+

∫ R4

0

∫
B(R)

|u||∆2(ψq
R)| dx dt

:= J1
R + J2

R.

(3.23)

Using (3.15) and Hölder inequality, because q ≥ p′, we write

J1
R ≲ R−4

∫ R4

R4

2

∫
B(R)

|u|ψq−1
R dx dt

≲ R
τ
p−4

∫ R4

R4

2

∫
B(R)

|x|−τ/p|u|ψ
q
p

R dx dt

≲ R
τ
p−4

(∫ R4

R4

2

∫
B(R)

|x|−τ |u|pψq
R dx dt

)1/p∣∣[R4

2
, R4]×B(R)

∣∣1− 1
p

≲ R(4+N)(1− 1
p )+

τ
p−4

(∫ R4

R4

2

∫
B(R)

|x|−τ |u|pψq
R dx dt

)1/p

.

(3.24)



8 T. SAANOUNI EJDE-2024/49

Furthermore, with (3.16) and (3.17), we obtain

J2
R

=

∫ R4

0

∫
B(R)

|u||∆2(ψq
R)| dx dt

≲ R−4

∫ R4

0

|ηR|q
∫
B(R)

|u|
(
ϕq−1
R |(∆2ϕ)(

x

R
)|+ ϕq−2

R |∇ϕ( x
R
)||∇(∆ϕ)(

x

R
)|

+ ϕq−2
R |(∆ϕ)( x

R
)|2 + ϕq−3

R |(∆ϕ)( x
R
)||∇ϕ( x

R
)|2 + ϕq−4

R |∇ϕ( x
R
)|4

)
dx dt

≲ R−4

∫ R4

0

∫
C(R

2 ,R)

|u|ψq−3
R dx dt.

(3.25)

Since qp′ ≥ 3, by (3.25) via Hölder inequality and arguing as in (3.24), it follows
that

J2
R ≲ R(4+N)(1− 1

p )+
τ
p−4

(∫ R4

0

∫
C(R

2 ,R)

|x|−τ |u|pψq
R dx dt

)1/p

. (3.26)

Now, by (3.23), (3.24) and (3.26), we obtain

JR ≲ R(4+N)(1− 1
p )+

τ
p−4J

1/p
R . (3.27)

Since (4 +N)(1− 1
p ) +

τ
p − 4 ≤ 0 because p ≤ 1 + 4−τ

N , then (3.27) implies that

1 ≳ lim
R→∞

JR = ℜ
(
λ

∫ ∞

0

∫
RN

|x|−τ |u|p dx dt
)
. (3.28)

Thus, |x|−τ/pu ∈ Lp([0,∞)× RN ) and so

lim
R→∞

J1
R = lim

R→∞
J2
R = 0. (3.29)

So, (3.23) via (3.29) gives

lim
R→∞

JR = 0. (3.30)

Finally, (3.30) implies that u = 0 and the proof is complete.

4. No global energy solutions

This section proves Theorem 2.6. The proof is reduced to the next Lemma via
Proposition 2.7.

Lemma 4.1. Let p > 1, ν > 0, 0 < τ < min{4, N} and an integer n ≥ 1 + 3p′.
Take u0 ∈ L1

loc(RN ) such that ℜ(u0) = 0 and u be a weak solution to (1.1) on

[0, T+
ν ). Then, there is C > 0 such that for any 0 < R < (T+

ν )1/4,

iν

∫
RN

u0ϕ
n
R dx ≤ CRN− 4−τ

p−1 . (4.1)

Moreover, if for some 0 < b < min{N, 4−τ
p−1}, it holds

ℑ(u0) ≤ −|x|−bχB(1), (4.2)



EJDE-2024/49 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 9

then, for any 0 < R < (T+
ν )1/4,

ν ≤ C
R−

(
4−τ
p−1−b

)∫
B( 1

R )
|x|−bϕn dx

. (4.3)

Furthermore, there is a ν0 > 0 such that

T+
ν ≤ CN,p,τ,bν

− 4
4−τ
p−1

−b , ∀ν > ν0. (4.4)

Proof. For some integer number n ≥ 1 + 3p′, let us define

In := In(R) :=

∫ R4

0

∫
B(R)

|x|−τ |u|pψn
R dx dt, (4.5)

Jn := Jn(R) :=

∫
B(R)

u0ϕ
n
R dx. (4.6)

Taking account of (2.5), we write

ℜ
(
iνJn + In

)
= ℜ

(
− i

∫ R4

0

∫
B(R)

u∂t
(
ψn
R

)
dx dt+

∫ R4

0

∫
B(R)

u∆2
(
ψn
R

)
dx dt

)
=

∫ R4

0

∫
B(R)

ℑu∂t
(
ψn
R

)
dx dt+

∫ R4

0

∫
B(R)

ℜu∆2
(
ψn
R

)
dx dt

:= (A) + (B).

(4.7)

By (3.17) and n ≥ p′, we have

(A) ≲ R−4

∫ R4

0

∫
B(R)

|u|ψn−1
R dx dt

≲ R−4

∫ R4

0

∫
B(R)

|u|ψn/p
R dx dt

≲ R
τ
p−4

∫ R4

0

∫
B(R)

|x|−τ/p|u|ψn/p
R dx dt.

(4.8)

So, with Hölder inequality via (4.8), we write

(A) ≲ R
τ
p−4I1/pn |[0, R4]×B(R)|1−

1
p ≲ R(4+N)(1− 1

p )+
τ
p−4I1/pn . (4.9)

By (3.15), (3.16) and (3.17), we have

(B)

≤
∫ R4

0

∫
B(R)

|u||∆2(ψn
R)| dx dt

≲ R−4

∫ R4

0

|ηR|n
∫
B(R)

|u|
(
ϕn−1
R |(∆2ϕ)(

x

R
)|+ ϕn−2

R |∇ϕ( x
R
)||∇(∆ϕ)(

x

R
)|

+ ϕn−2
R |(∆ϕ)( x

R
)|2 + ϕn−3

R |(∆ϕ)( x
R
)||∇ϕ( x

R
)|2 + ϕn−4

R |∇ϕ( x
R
)|4

)
dx dt

≲ R−4

∫ R4

0

∫
C(R

2 ,R)

|u|ψn−3
R dx dt.

(4.10)



10 T. SAANOUNI EJDE-2024/49

Since n ≥ 3p′, by (4.10) via Hölder inequality and arguing as in (4.9), it follows
that

(B) ≲ R(4+N)(1− 1
p )+

τ
p−4

(∫ R4

0

∫
C(R

2 ,R)

|x|−τ |u|pψn
R dx dt

)1/p

. (4.11)

So, (4.11), (4.9) and (4.7), via Young inequality give

−νℑJn = ℜ
(
iνJn

)
≤ CR(4+N)(1− 1

p )+
τ
p−4I1/pn − In

≤ 1

p′
(
CR(4+N)(1− 1

p )+
τ
p−4p−

1
p )p

′
+ In − In

≲ R4+N+ τ
p−1−4p′

.

(4.12)

This proves (4.1). By (4.6) and (4.2), we obtain

−ℑJn = −
∫
RN

ℑu0ϕnR dx

= −RN

∫
RN

ℑu0(Rx)ϕn dx

≥ RN−b

∫
B( 1

R )

|x|−bϕn dx.

(4.13)

Thus, (4.1) and (4.13) give

νRN−b

∫
B( 1

R )

|x|−bϕn dx ≲ −νℑJn = iν

∫
RN

u0ϕ
n
R dx ≤ CR4+N+ τ

p−1−4p′
. (4.14)

This proves (4.3). Now, there is a ν0 > 0 such that for any ν > ν0, we have
T+
ν ≤ 16. Indeed, otherwise we take R = 2 in (4.3) and we obtain because b < N ,

the contradiction

ν ≤ C
24+b+ τ

p−1−4p′∫
B( 1

2 )
|x|−bϕn dx

= C
24+b+ τ

p−1−4p′∫
B( 1

2 )
|x|−b dx

≤ C24+b+ τ
p−1−4p′

:= ν0.

(4.15)

So, taking ν > ν0 and 0 < R < (T+
ν )1/4 ≤ 2, we write by (4.3),

ν ≤ C
R4+b+ τ

p−1−4p′∫
B( 1

R )
|x|−bϕn dx

= C
R4+b+ τ

p−1−4p′∫
B( 1

2 )
|x|−b dx

≲ R4+b+ τ
p−1−4p′

.

(4.16)

Now, since b < 4−τ
p−1 , (4.16) gives 4 + b+ τ

p−1 − 4p′ < 0 and so

T+
ν ≲ ν

− 4
4p′−4−b− τ

p−1 = ν
− 4

4−τ
p−1

−b . (4.17)

This establishes (4.4) and completes the proof. □



EJDE-2024/49 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 11

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Tarek Saanouni

Department of Mathematics, College of Science, Qassim University, Buraydah, King-
dom of Saudi Arabia

Email address: t.saanouni@qu.edu.sa


	1. Introduction
	2. Background and main result
	2.1. Preliminaries
	2.2. Main results
	2.3. Sketch of the proofs

	3. No global mass solutions
	3.1. Weak solutions 
	3.2. Vanishing solutions

	4. No global energy solutions
	References