Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 55, pp. 1–21.

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

DOI: 10.58997/ejde.2025.55

NON GLOBAL SOLUTIONS FOR NON-RADIAL INHOMOGENEOUS

NONLINEAR SCHRÖDINGER EQUATIONS

RUOBING BAI, TAREK SAANOUNI

Abstract. This work concerns the inhomogeneous Schrödinger equation

i∂tu−Ks,λu+ F (x, u) = 0, u(t, x) : R× RN → C.

Here, s ∈ {1, 2}, N > 2s and λ > −(N − 2)2/4. The linear Schrödinger operator is Ks,λ :=

(−∆)s + (2− s) λ
|x|2 , and the focusing source term can be local or non-local

F (x, u) ∈ {|x|−2τ |u|2(q−1)u, |x|−τ |u|p−2
(
Jα ∗ | · |−τ |u|p

)
u}.

The Riesz potential is Jα(x) = CN,α|x|−(N−α), for certain 0 < α < N . The singular decaying

term |x|−2τ , for some τ > 0 gives an inhomogeneous non-linearity. One considers the inter-

critical regime, namely 1 +
2(s−τ)

N
< q < 1 +

2(s−τ)
N−2s

and 1 +
2(s−τ)+α

N
< p < 1 +

2(s−τ)+α
N−2s

.

The purpose is to prove the finite time blow-up of solutions with datum in the energy space, not

necessarily radial or with finite variance. The assumption on the data is expressed in two different
ways. The first one is in the spirit of the potential well method due to Payne-Sattinger. The

second one is the ground state threshold standard condition. The proof is based on Morawetz
estimates and a non-global ordinary differential inequality. This work complements the recent

paper by Bai and Li [4] in many directions.

1. Introduction

This article concerns the Cauchy problem for an inhomogeneous generalized Hartree equation

i∂tu−Ks,λu+ |x|−τ |u|p−2
(
Jα ∗ | · |−τ |u|p

)
u = 0;

u(0, ·) = u0,
(1.1)

and the Cauchy problem for an inhomogeneous Schrödinger equation

i∂tu−Ks,λu+ |x|−2τ |u|2(q−1)u = 0;

u(0, ·) = u0.
(1.2)

Hereafter, N
2 > s ∈ {1, 2} and u = u(t, x) : R × RN → C. The linear Schrödinger operator is

denoted by Ks,λ := (−∆)s + (2 − s) λ
|x|2 . We considered 2 cases: The first one is Kλ := K1,λ =

−∆+ λ
|x|2 , which corresponds to Schrödinger equation with inverse square potential. The second

one is K2,λ := ∆2, which corresponds to fourth-order Schrödinger equation. The inhomogeneous
singular decaying term is | · |−2τ for some τ > 0. The Riesz-potential is defined on RN by

Jα :=
Γ(N−α

2 )

Γ(α2 )π
N/22α

| · |α−N , 0 < α < N.

In all this expression, one assumes that

min{τ, α,N − α,N − τ, 2− 2τ + α} > 0. (1.3)

2020 Mathematics Subject Classification. 35Q55.
Key words and phrases. Inhomogeneous Schrödinger problem; nonlinear equations; bi-harmonic;

inverse square potential; finite time blow-up.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted March 7, 2025. Published May 26, 2025.

1



2 R. BAI, T. SAANOUNI EJDE-2025/55

Motivated by the sharp Hardy inequality [5],

(N − 2)2

4

∫
RN

|f(x)|2

|x|2
dx ≤

∫
RN

|∇f(x)|2 dx, (1.4)

one assumes that λ > −(N − 2)2/4, which guarantees that extension of −∆ + λ
|x|2 , denoted by

Kλ is a positive operator. In the range − (N−2)2

4 < λ < 1 − (N−2)2

4 , the extension is not unique
[21, 42]. In such a case, one picks the Friedrichs extension [21, 33].

Note that by the definition of the operator Kλ and Hardy estimate (1.4), one has

∥
√
Kλ · ∥ =

(
∥∇ · ∥2 + λ∥ ·

|x|
∥2
)1/2

≃ ∥ · ∥Ḣ1 . (1.5)

The nonlinear equations of Schrödinger type (1.1) and (1.2) model many physical phenomena.
For s = 1, they are used in nonlinear optical systems with spatially dependent interactions [6].
In particular, when λ = 0, they can be thought of as modeling inhomogeneities in the medium in
which the wave propagates [24]. When τ = 0, they model a quantum field equations or black hole
solutions of the Einstein’s equations [21]. For s = 2, the above equations are called fourth-order
Schrödinger equations. The bi-harmonic Schrödinger problem was considered first in [22, 23] to
take 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. The source term can be understood as
a nonlinear potential affected by electron density [7].

The literature dealing with (1.1) and (1.2) is copious, and naturally some references are missing
here. Let us start with the Schrödinger equation with inverse square potential, which corresponds
to s = 1. Using the energy method, [40, 41] investigated the local well-posedness in the energy
space. Moreover, the local solution extends globally in time if either defocusing case or focusing,
mass-subcritical case. Later on, [9] revisits the same problem, where the authors studied the
local well-posedness and small data global well-posedness in the energy-sub-critical case by using
the standard Strichartz estimates combined with the fixed point argument. See also [2] for the
ground state threshold of global existence versus blow-up dichotomy in the inter-critical regime.
Furthermore, [9] showed a scattering criterion and constructed a wave operator for the inter-
critical case. The well-posedness and blow-up in the energy critical regime were investigated in
[20]. The inhomogeneous generalized Hartree equation was treated first by the author [1], where
the ground state threshold dichotomy was investigated using a sharp adapted Gagliargo-Nirenberg
type estimate. After that, the second author treated the intermediate case in the sense that (1.1)

is locally well-posed in Ḣ1 ∩ Ḣsc , 0 < sc < 1, but this does not imply the inter-critical case Hsc .
The scattering under the ground state threshold with spherically symmetric data, was proved by
the second author [39]. The scattering was extended to the non-radial regime in [43]. The well-
posedness in the energy-critical regime was investigated recently [26, 25]. To this end, the authors
approach to the matter based on the Sobolev-Lorentz space which can lead to perform a finer
analysis. This is because it makes it possible to control the non-linearity involving the singularity
|x|−τ as well as the Riesz potential more effectively. Now, one deals with the bi-harmonic case,
namely s = 2. For a local source term, in [17], the local well-posedness was obtained in the energy
sub-critical regime. This result was improved in [3]. The scattering was investigated in [18, 10, 14].
For a non-local source term, the local existence of energy solutions and the scattering were proved
by the second author in [34, 36]. See also [37, 38] for the energy-critical regime.

The finite time concentration of energy solutions to non-linear Schrödinger equations has a
long history. Indeed, in the mass-super-critical focusing regime, it is known that an energy data
with finite variance or which is radial gives a blowing up solution for negative energy [16, 30]. A
similar result for non-radial data and with infinite variance is open except for N = 1, see [31].
The results of blow-up in some other situations can be referred to [15, 19, 28, 29] and references
therein. Recently, some works try to remove the radial or finite variance data assumption in the
inhomogeneous case. Indeed, the second author proved in [4] the finite time blow-up of energy
solutions under the ground state threshold in a restricted range of the source term exponent.
In the mass-critical regime, the blow-up of energy solutions with negative energy was obtained
recently [11].



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 3

The blow-up of energy solutions to bi-harmonic Schrödinger equations was open for a long
time because of the lack of a variance identity. Many authors investigated the blow-up of radial
solutions, since the pioneering work [8] using a localized virial identity for radial datum. See, for
instance [12, 36]. Recently the blow-up for arbitrary datum with negative energy, in the energy
space, was obtained in [13] for a perturbed bi-harmonic NLS. This result don’t extend to (1.2) for
s = 2.

The purpose of this article is to investigate the finite time blow-up of energy solutions to the
Schrödinger problems (1.1) and (1.2). The novelty is to prove the non-global existence of solutions
with arbitrary negative energy datum. Precisely, one don’t require any radial or finite variance
assumption for the datum. In the case s = 1, this work complements the paper of the second
author [4] for λ ̸= 0 and for a non-local source term. Moreover, one considers a weaker assumption
on the datum. In the case s = 2, this work complements the paper [13] to the inter-critical regime,
and for a non-local source term. Furthermore, this work gives a natural complement of the paper
[34], where the first author deals with the scattering of the bi-harmonic Schrödinger equation in
the inter-critical focusing regime under the ground state threshold.

The rest of this article is organized as follows. The next section contains the main results and
some useful estimates. Sections 3 and 4 contain the proofs of the main results.

2. Background and main results

This section contains the main results and some useful estimates.

2.1. Preliminaries. Here and hereafter, one denotes for simplicity some standard Lebesgue and
Sobolev spaces and norms as follows

Lr := Lr(RN ), W s,r :=W s,r(RN ), Hs :=W s,2, ∥ · ∥r := ∥ · ∥Lr , ∥ · ∥ := ∥ · ∥2.
Let us also define the real numbers

B :=
Np−N − α+ 2τ

s
, A := 2p−B, B′ :=

Nq −N + 2τ

s
, A′ := 2q −B′.

If u ∈ Hs, one defines the quantities related to energy solutions of (1.1) and (1.2),

P[u] :=

∫
RN

|x|−τ
(
Jα ∗ | · |−τ |u|p

)
|u|p dx, Q[u] :=

∫
RN

|x|−2τ |u|2q dx,

I[u] := ∥
√
Ks,λu∥2 −

B

2p
P[u], J [u] := ∥

√
Ks,λu∥2 −

B′

2q
Q[u];

M[u] :=

∫
RN

|u(x)|2 dx, E [u] := ∥
√

Ks,λu∥2 −
1

p
P[u],

E ′[u] := ∥
√

Ks,λu∥2 −
1

q
Q[u].

We denote also the so-called actions

S[u] := E [u] +M[u], (2.1)

S ′[u] := E ′[u] +M[u]. (2.2)

Take also the real numbers

m := inf
0 ̸=u∈Hs

{
S[u] : I[u] = 0

}
; (2.3)

m′ := inf
0̸=u∈Hs

{
S ′[u] : J [u] = 0

}
. (2.4)

Finally, we define the sets, which are non-empty with a scaling argument

A− :=
{
u ∈ Hs : S[u] < m : I[u] < 0

}
, (2.5)

A′− :=
{
u ∈ Hs : S ′[u] < m′, J [u] < 0

}
. (2.6)

Then equation (1.1) has the scaling invariance

uκ := κ
2s−2τ+α
2(p−1) u(κ2s·, κ·), κ > 0. (2.7)



4 R. BAI, T. SAANOUNI EJDE-2025/55

The critical exponent sc keeps invariant the homogeneous Sobolev norm

∥uκ(t)∥Ḣµ = κµ−(N
2 − 2s−2τ+α

2(p−1)
)∥u(κ2st)∥Ḣµ := κµ−sc∥u(κ2st)∥Ḣµ .

Two cases are of particular interest in the physical context. The first one sc = 0 corresponds to
the mass-critical case which is equivalent to p = pc := 1 + 2s−2τ+α

N . This case is related to the
conservation of the mass M given above. The second one is the energy-critical case sc = s, which
corresponds to p = pc := 1 + 2s−2τ+α

N−2s . This case is related to the conservation of the energy E
defined above. A particular periodic global solution of (1.1) takes the form eitφ, where φ satisfies

Ks,λφ+ φ = |x|−τ |φ|p−2
(
Jα ∗ | · |−τ |φ|p

)
φ, 0 ̸= φ ∈ Hs. (2.8)

The equation (1.2) has the scaling invariance

uκ := κ
s−τ
q−1 u(κ2s·, κ·), κ > 0. (2.9)

The critical exponent s′c keeps invariant the following homogeneous Sobolev norm

∥uκ(t)∥Ḣµ = κµ−(N
2 − s−τ

q−1 )∥u(κ2st)∥Ḣµ := κµ−s′c∥u(κ2st)∥Ḣµ .

Two cases are of particular interest in the physical context. The first one s′c = 0 corresponds
to the mass-critical case which is equivalent to q = qc := 1 + 2s−2τ

N . This case is related to the
conservation of the mass. The second one is the energy-critical case s′c = s, which corresponds to
q = qc := 1 + 2s−2τ

N−2s . This case is related to the conservation of the energy E ′ defined above. A

particular periodic global solution of (1.2) takes the form eitψ, where ψ satisfies

Ks,λψ + ψ = |x|−2τ |ψ|2(q−1)ψ, 0 ̸= ψ ∈ Hs. (2.10)

The existence of such a ground state is related to the next Gagliardo-Nirenberg type inequalities
[36, 35].

Proposition 2.1. Let s ∈ {1, 2}, N > 2s, 0 < α < N and 1 + α
N < p < pc. If s = 1, one

assumes that λ > − (N−2)2

4 and (1.3) holds. Moreover, if s = 2, one assumes that 0 < 2τ <
min{N + α, 4(1 + α

N )}. Thus,

(1) There exists a sharp constant CN,p,τ,α,λ > 0 such that for all u ∈ Hs,∫
RN

|x|−τ |u|p
(
Jα ∗ | · |−τ |u|p

)
dx ≤ CN,p,τ,α,λ∥u∥A∥

√
Ks,λu∥B ; (2.11)

(2) there exists φ a solution to (2.8) satisfying

CN,p,τ,α,λ =
2p

A

(A
B

)B/2 1

∥φ∥2(p−1)
; (2.12)

(3) one has the following Pohozaev identities

P[φ] =
2p

A
M[φ] =

2p

B
∥
√

Ks,λφ∥2. (2.13)

The next Gagliardo-Nirenberg type inequality [36, 9] is essential to estimate an eventual solution
to the problem (1.2).

Proposition 2.2. Let s ∈ {1, 2}, N > 2s, λ > − (N−2)2

4 , 0 < τ < s and 1 < q < qc. Thus,

(1) there exists a sharp constant CN,q,τ,λ > 0, such that for all v ∈ Hs,∫
RN

|x|−2τ |v|2q dx ≤ CN,q,τ,λ∥v∥A
′
∥
√

Ks,λv∥B
′

(2) there exists ψ a solution to (2.10) satisfying

C(N, q, τ) =
2q

A′ (
A′

B′ )
B′
2 ∥ψ∥−2(q−1), (2.14)

where ψ is a solution to (2.10);
(3) one has the following Pohozaev identities

Q[ψ] =
2q

A′M[ψ] =
2q

B′ ∥
√

Ks,λψ∥2. (2.15)



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 5

In the inter-critical regime 0 < sc < s, one denotes the positive real number s
sc
−1 := αc ∈ (0, 1),

φ be a ground state of (2.8) and the scale invariant quantities

ME [u0] :=
(M[u0]

M[φ]

)αc
(E [u0]
E [φ]

)
, MG[u0] :=

(∥u0∥
∥φ∥

)αc
(∥√Ks,λu0∥
∥
√

Ks,λφ∥

)
.

Similarly, in the inter-critical regime 0 < s′c < s, one denotes the positive real number s
s′c

− 1 :=

α′
c ∈ (0, 1), ψ be a ground state of (2.10) and the scale invariant quantities

ME ′[u0] :=
(M[u0]

M[ψ]

)α′
c
(E ′[u0]

E ′[ψ]

)
, MG′[u0] :=

(∥u0∥
∥ψ∥

)α′
c
(∥√Ks,λu0∥
∥
√

Ks,λψ∥

)
.

In the next sub-section, one lists the contributions of this note.

2.2. Main results. First, one deals with the non-global existence of energy solutions to the
generalized Hartree problem (1.1).

Theorem 2.3. Let s ∈ {1, 2}, N > 2s, 0 < α < N , λ ≥ 0 and 0 < τ < sα+N
N such that (1.3)

holds. Suppose that max{2, pc} < p < pc and p ≤ 1 + 2s+α−τ
N . Take φ be a ground state solution

to (2.8) and u ∈ CT∗(Hs) be a maximal solution of the focusing problem (1.1). Thus, u blows-up
in finite time if one of the following assumptions holds

u0 ∈ A−, (2.16)

MG[u0] > 1 >ME [u0]. (2.17)

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

• In [34], the local existence of energy solutions for (1.1) with s = 2 was proved under the
supplementary assumption 0 < 2τ < min{4(1 + α

N ),−N + 8 + α}. Moreover, in [35], the
local existence of energy solutions for (1.1) with s = 1 was proved under the supplementary
assumption 1 + α− 2τ > 0.

• The space A− is proved to be stable under the flow of (1.1).
• The first part of the Theorem follows the potential well theory due to Payne-Sattinger
[32].

• The assumption on the source term exponent, can be written as 2 < B ≤ 2 + τ
s .

• The slab (pc, 1 +
2s+α−τ

N ] has a length of τ
N , which is independent of s.

• The restriction λ ≥ 0 is needed in the proof.
• The above result doesn’t extend to the limiting case τ = 0, which is still an open problem.
This gives an essential difference between the NLS and the INLS.

• In a paper in progress, the authors treat the finite time blow-up of energy solutions in the
mass-critical bi-harmonic regime.

Second, one deals with the non-global existence of energy solutions to the Schrödinger problem
(1.2).

Theorem 2.4. Let s ∈ {1, 2}, N > 2s, λ ≥ 0 and 0 < τ < 2. Assume that qc < q < qc and

q ≤ 1 + 2s+2τ(s−1)
N . Take ψ be a ground state solution to (2.10) and u ∈ CT∗(Hs) be a maximal

solution of the focusing problem (1.2). Thus, u blows-up in finite time if one of the following
assumptions holds

u0 ∈ A′−, (2.18)

MG′[u0] > 1 >ME ′[u0]. (2.19)

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

• In [9, 40], the local existence of energy solutions to (1.2) for s = 1 was proved under the
supplementary assumption τ < 1. In [9, 40, 17], the local existence of energy solutions to
(1.2) for s = 2 was proved under the supplementary assumption q > 1 + 1−2τ

N .
• Assumption (2.19) is used to prove that (2.18) is stable under the flow of (1.2).
• In [4], the first author proved the finite time blow-up of energy solutions for (1.2) for s = 1
and λ = 0 under the assumption (2.19).



6 R. BAI, T. SAANOUNI EJDE-2025/55

2.3. Useful estimates. In this sub-section, one gives some standard tools needed in the sequel.
Let us start with Hardy-Littlewood-Sobolev inequality [27].

Lemma 2.5. Let N ≥ 1 and 0 < α < N .

(1) Let r > 1 such that 1
r = 1

s + α
N . Then,

∥Jα ∗ g∥s ≤ CN,s,α∥g∥r, ∀g ∈ Lr.

(2) Let 1 < s, r <∞ be such that 1
r + 1

s = 1
t +

α
N . Then

∥f(Jα ∗ g)∥t ≤ CN,s,α∥f∥r∥g∥s, ∀(f, g) ∈ Lr × Ls.

Let ξ : RN → R be a convex smooth function. We define the variance potential

Vξ :=

∫
RN

ξ(x)|u(·, x)|2 dx, (2.20)

and the Morawetz action

Mξ = 2ℑ
∫
RN

ū(∇ξ · ∇u) dx := 2ℑ
∫
RN

ū(ξjuj) dx, (2.21)

where here and sequel, repeated indices are summed. Let us give a Morawetz type estimate for
the Schrödinger equation with inverse square potential [2].

Proposition 2.6. Take u, v ∈ CT∗(H1) be the local solutions to (1.1) and (1.2) for s = 1,
respectively. Let ξ : RN → R be a smooth function. Then, the following equality holds on [0, T ∗),

V ′′
ξ [u] =M ′

ξ[u]

= 4

∫
RN

∂l∂kξℜ(∂ku∂lū) dx−
∫
RN

∆2ξ|u|2 dx+ 4λ

∫
RN

∇ξ · x |u|
2

|x|4
dx

+ 2(
2

p
− 1)

∫
RN

∆ξ|x|−τ |u|p(Jα ∗ | · |−τ |u|p) dx+
4

p

∫
RN

∇ξ · ∇(|x|−τ )|u|p
(
Jα ∗ | · |−τ |u|p

)
dx

+
4

p
(α−N)

∫
RN

|x|−τ |u|p∇ξ( ·
| · |2

Jα ∗ | · |−τ |u|p) dx.

Moreover,

V ′′
ξ [v] =M ′

ξ[v] = 4

∫
RN

∂l∂kξℜ(∂kv∂lv̄) dx−
∫
RN

∆2ξ|v|2 dx+ 4λ

∫
RN

∇ξ · x |v|
2

|x|4
dx

+ 2(
1

q
− 1)

∫
RN

∆ξ|x|−2τ |v|2q dx+
2

q

∫
RN

∇ξ · ∇(|x|−2τ )|v|2q dx.

Finally, one gives a Morawetz estimate for the bi-harmonic Schrödinger equation [34].

Proposition 2.7. Take u, v ∈ CT∗(H2) be the local solutions to (1.1) and (1.2) for s = 2,
respectively. Let ξ : RN → R be a smooth function. Then, the following equalities hold on [0, T ∗),

M ′
ξ[u] = −2

∫
RN

(
2∂jk∆ξ∂ju∂kū− 1

2
(∆3ξ)|u|2 − 4∂jkξ∂iku∂ij ū+∆2ξ|∇u|2

)
dx

− 2
(
(1− 2

p
)

∫
RN

∆ξ(Jα ∗ | · |−τ |u|p)|x|−τ |u|p dx

− 2

p

∫
RN

∂kξ∂k(|x|−τ [Jα ∗ | · |−τ |u|p])|u|p dx
)
,

(2.22)

M ′
ξ[v] = −2

∫
RN

(
2∂jk∆ξ∂jv∂kv̄ −

1

2
(∆3ξ)|v|2 − 4∂jkξ∂ikv∂ij v̄

+∆2ξ|∇v|2 + q − 1

q
(∆ξ)|x|−2τ |v|2q − 1

q
∇ξ · ∇(|x|−2τ )|v|2q

)
dx

. (2.23)

The next radial identities will be useful in the sequel.

∂2

∂xl∂xk
:= ∂l∂k =

(δlk
r

− xlxk
r3

)
∂r +

xlxk
r2

∂2r , (2.24)



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 7

∆ = ∂2r +
N − 1

r
∂r, (2.25)

∇ =
x

r
∂r. (2.26)

In the rest of this note, one takes a smooth radial function ξ(x) := ξ(|x|) such that

ξ : r →

{
r2, if 0 ≤ r ≤ 1;

0, if r ≥ 10.

So, on the unit ball of RN , one has

ξij = 2δij , ∆ξ = 2N, ∂γξ = 0 for |γ| ≥ 3.

Now, for R > 0, via (2.21), one takes

ξR := R2ξ(
| · |
R

) and MR :=MξR .

By [13, Lemma 2.1], one can impose that

max{ξ
′
R

r
− 2, ξ′′R − ξ′R

r
} ≤ 0. (2.27)

From now one hides the time variable t for simplicity, displaying it out only when necessary. More-
over, one denotes the centered ball of RN with radius R > 0 and its complementary, respectively
by B(R) and Bc(R). In what follows, one proves the main results of this note.

3. Schrödinger equation with non-local source term

In this section, we establish Theorem 2.3.

3.1. Schrödinger equation with inverse square potential. In this sub-section, one takes
s = 1.

First case. Assume that (2.16) holds. We start with the next auxiliary result.

Lemma 3.1. (1) The set A− is stable under the flow of (1.1).
(2) There exists ε > 0, such that for any t ∈ [0, T ∗),

I[u(t)] + ε∥
√

Kλu(t)∥2 ≤ −B
4

(
m− S[u(t)]

)
. (3.1)

Proof. (1) Assume that u0 ∈ A− and that there is 0 < t0 < T ∗ such that u(t0) /∈ A−. This
implies that I[u(t0)] ≥ 0 and by a continuity argument, there is 0 < t1 such that I[u(t1)] = 0 and
S[u(t1)] < m. This contradicts the definition of m and proves the first point of Lemma 3.1.

(2) Now, taking the scaling uρ := ρ
N
2 u(ρ·) for ρ > 0, we compute

∥uρ∥ = ∥u∥; (3.2)

∥
√
Kλuρ∥ = ρ∥

√
Kλu∥; (3.3)

P[uρ] = ρBP[u]. (3.4)

Moreover, take the real function 𭟋 : ρ 7→ S[uρ], we obtain 𭟋(ρ) = ρ2∥
√
Kλu∥2 + ∥u∥2 − ρB

p P[u]

and the first derivative reads

𭟋′(ρ) = 2ρ∥
√
Kλu∥2 −B

ρB−1

p
P[u] = 2ρ−1I[uρ]. (3.5)

Hence, this implies

ρ𭟋′(ρ) = 2ρ2∥
√

Kλu∥2 −B
ρB

p
P[u] = 2I[uρ]. (3.6)

Moreover, since B > 2, we obtain(
ρ𭟋′(ρ)

)′
= 4ρ∥

√
Kλu∥2 −B2 ρ

B−1

p
P[u] = B𭟋′(ρ)− 2(B − 2)ρ∥

√
Kλu∥2 ≤ B𭟋′(ρ). (3.7)



8 R. BAI, T. SAANOUNI EJDE-2025/55

Now, we claim that there exists ρ0 ∈ (0, 1) such that

I[uρ0
] = 0. (3.8)

Indeed, by (3.3) and (3.4), we have

I[uρ] = ρ2
(
∥
√
Kλu∥2 −

ρB−2

2p
P[u]

)
:= ρ2ℵ(ρ). (3.9)

Note that ℵ(0) > 0 and ℵ(1) = I[u] < 0, thus there exists ρ0 ∈ (0, 1) such that ℵ(ρ0) = 0,
the claim is proved. Thus, by (3.5), we have 𭟋′(ρ0) = 0 and 𭟋(ρ0) = S[uρ0

] ≥ m. Hence, an
integration of (3.7) on [ρ0, 1] gives

𭟋′(1)− ρ0𭟋′(ρ0) ≤ B𭟋(1)−B𭟋(ρ0).

Note that 𭟋′(1) = 2I[u], ρ0𭟋′(ρ0) = 2I[uρ0
] = 0, and 𭟋(1) = S[u], the above inequality further

implies

I[u] ≤ −B
2

(
m− S[u]

)
. (3.10)

On the other hand, we write

∥
√
Kλu∥2 =

B

B − 2

(
S[u]− 2

B
I[u]− ∥u∥2

)
. (3.11)

Hence, by (3.10), there exists 0 < ε≪ 1, such that

I[u] + ε∥
√
Kλu∥2 =

(
1− 2ε

B − 2

)
I[u] + ε

B

B − 2

(
S[u]− ∥u∥2

)
≤ −B

2

(
1− 2ε

B − 2

)(
m− S[u]

)
+ ε

B

B − 2
S[u]

≤ −B
4

(
m− S[u]

) . (3.12)

The last statement of Lemma 3.1 is proved by (3.12). □

Now we turn to the proof of the main results. Taking into account Proposition 2.6, one has
M ′

R := (L) + (N), where

(L) = −
∫
RN

∆2ξR|u|2 dx+ 4

∫
RN

∂l∂kξRℜ(∂ku∂lū) dx+ 4λ

∫
RN

∇ξR · x |u|
2

|x|4
dx,

and

(N) = 2(
2

p
− 1)

∫
RN

∆ξR|x|−τ |u|p(Jα ∗ | · |−τ |u|p) dx

− 4τ

p

∫
RN

x · ∇ξR|x|−τ−2|u|p
(
Jα ∗ | · |−τ |u|p

)
dx

+
4

p
(α−N)

∫
RN

|x|−τ |u|p∇ξR
( ·
| · |2

Jα ∗ | · |−τ |u|p
)
dx

:= (N)1 + (N)2 + (N)3.

For the term (L), with the properties of ξR, namely (2.27) and the radial identities, it follows that

(L) = −
∫
RN

∆2ξR|u|2 dx+ 4

∫
RN

|∇u|2 ξ
′
R

r
dx+ 4

∫
RN

|x · ∇u|2(ξ
′′
R

r2
− ξ′R
r3

) dx

+ 4λ

∫
RN

|u|2

r3
ξ′R dx

≤ −
∫
RN

∆2ξR|u|2 dx+ 8

∫
RN

|∇u|2 dx+ 8λ

∫
RN

|u|2

r2
dx.

(3.13)



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 9

For the terms (N)1 and (N)2 in (N), taking account of the truncation function properties and the
conservation laws, one has

(N)1 + (N)2 = 2(
4N − 4τ

p
− 2N)P[u] +O

(∫
Bc(R)

|x|−τ |u|p
(
Jα ∗ | · |−τ |u|p

)
dx

)
.

For the third term (N)3 in (N), with the calculations done in [34], one has

(N)3 =
4(α−N)

p

∫
B(R)×B(R)

Jα(x− y)|y|−τ |u(y)|p|x|−τ |u(x)|p dx dy

+O
(∫

Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
=

4(α−N)

p

∫
B(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u(x)|p dx

+O
(∫

Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
=

4(α−N)

p
P[u] +O

(∫
Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
.

(3.14)

Collecting the above estimates, one obtains

(N) = −4B

p
P[u] +O

(∫
Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
. (3.15)

Hence, by (3.13) and (3.15), one has

M ′
R ≤ −

∫
RN

∆2ξR|u|2 dx+ 8

∫
RN

|∇u|2 + 8λ

∫
RN

|u|2

r2
dx− 4B

p
P[u]

+O
(∫

Bc(R)

|x|−τ |u|p
(
Jα ∗ | · |−τ |u|p

)
dx

)
≤ 8

(
∥
√
Kλu∥2 −

B

2p
P[u]

)
+O

(∫
Bc(R)

|x|−τ |u|p
(
Jα ∗ | · |−τ |u|p

)
dx

)
+O(R−2).

(3.16)

Now, with Lemma 2.5, the Hölder and Gagliardo-Nirenberg inequalities via the mass conservation,
one writes∫

Bc(R)

|x|−τ |u|p
(
Jα ∗ | · |−τ |u|p

)
dx ≲ ∥|x|−τ |u|p∥ 2N

α+N
∥|x|−τ |u|p∥

L
2N

α+N (Bc(R))

≲ R−τ∥u∥p2Np
α+N

∥|x|−τ |u|p∥ 2N
α+N

≲ R−τ∥u0∥p−
N(p−1)−α

2 ∥∇u∥
N(p−1)−α

2 ∥|x|−τ |u|p∥ 2N
α+N

≲ R−τ∥∇u∥
N(p−1)−α

2 ∥|x|−τ |u|p∥ 2N
α+N

.

(3.17)

Now, using Proposition 2.2, with 2Nτ
α+N instead of 2τ and 2Np

α+N instead of 2q, via the fact that
0 < τ < 1 + α

N and p < pc, one has

∥ |x|−τ |u|p∥ 2N
α+N

=
(∫

RN

|x|−
2Nτ
α+N |u|

2Np
α+N dx

)α+N
2N

≲ ∥u0∥p−(
N(p−1)−α

2 +τ)∥
√

Kλu∥
N(p−1)−α

2 +τ .

(3.18)

Thus, recall that B = Np−N −α+2τ , by (3.16) via (3.18) and (1.5), one obtains for large R≫1,

M ′
R ≤ CI[u] + C

Rτ
∥
√

Kλu∥B−τ +
C

R2
. (3.19)

Since I[u] < 0, by Gagliardo-Nirenberg estimate in Proposition 2.1, via the mass conservation
law, one has

∥
√
Kλu∥2 ≲ P[u] ≲ ∥

√
Kλu∥B∥u∥2p−B ≲ ∥

√
Kλu∥B . (3.20)



10 R. BAI, T. SAANOUNI EJDE-2025/55

Thus, by B > 2, there exists C0 > 0 such that for any t ∈ [0, T ∗),

∥
√

Kλu(t)∥ ≥ C0. (3.21)

Hence, (3.1) and (3.19)-(3.12) give for 2 < B ≤ 2 + τ and R≫ 1,

M ′
R[u] ≲ I[u] +R−2 +R−τ∥

√
Kλu∥B−τ

≲ −∥
√
Kλu∥2 +R−2 +R−τ∥

√
Kλu∥B−τ

≲ ∥
√
Kλu∥2

(
− 1 +R−2 +R−τ∥

√
Kλu∥B−2−τ

)
≲ −∥

√
Kλu∥2.

(3.22)

Time integration, (3.21), and (3.22) imply that

MR[u(t)] ≲ −t, t > T > 0. (3.23)

By time integration again, from (3.22), it follows that

MR[u(t)] ≲ −
∫ t

T

∥
√

Kλu(s)∥2 ds. (3.24)

Now, the definition (2.21) via (1.5) gives

|MR| = 2|ℑ
∫
RN

ū(∇ξR · ∇u) dx| ≲ R∥∇u∥∥u∥ ≲ R∥
√
Kλu∥. (3.25)

Thus, by (3.23), (3.24) and (3.25), it follows that∫ t

T

∥
√
Kλu(s)∥2 ds ≲ |MR[u(t)]| ≲ R∥

√
Kλu(t)∥, ∀t > T. (3.26)

Take the real function f(t) :=
∫ t

T
∥
√
Kλu(s)∥2. By (3.26), one obtains f2 ≲ f ′. This ODI has no

global solution. Indeed, for T ′ > T > t, an integration gives

t− T ′ ≲
∫ t

T ′

f ′(s)

f2(s)
ds =

1

f(T ′)
− 1

f(t)
≤ 1

f(T ′)
.

This implies T ′ + c
f(T ′) . This completes the proof.

Second case. Assume that (2.17) holds. Taking account of the previous sub-section, it is sufficient
to prove the next result.

Lemma 3.2. There exist C > 0 and ε > 0, such that for any t ∈ [0, T ∗), the following statements
hold:

I[u(t)] < −C < 0,

and

I[u(t)] + ε∥
√

Kλu(t)∥2 < 0. (3.27)

Proof. (1) Define the quantity C :=
CN,p,τ,λ

p ∥u∥A. Then, by Proposition 2.1, one writes

F (∥
√
Kλu(t)∥2) := ∥

√
Kλu(t)∥2 − C∥

√
Kλu(t)∥B ≤ E [u0], on [0, T ∗). (3.28)

Now, since B > 2, the above real function has a maximum

F (x1) := F
[( 2

CB

) 2
B−2

]
=

( 2

CB

) 2
B−2

(
1− 2

B

)
.

Moreover, thanks to Pohozaev identities (2.13) and the condition (2.17), it follows that

E [u0] <
B − 2

A

(
M[u0]

)−αc
(
M[φ]

) 1
sc . (3.29)



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 11

In addition, by (2.12), one obtains

F (x1) =
( 2

BC

) 2
B−2

(
1− 2

B

)
=

(
(
A

B
)1−

B
2 (M[φ])p−1(M[u0])

−A
2

) 2
B−2

(
1− 2

B

)
=
B − 2

A

(
(M[u0])

−A
2 (M[φ])p−1

) 2
B−2

=
B − 2

A

(
M[u0]

)−αc
(
M[φ]

) 1
sc .

(3.30)

Relations (3.29) and (3.30) imply that E [u0] < F (x1). By the previous inequality and (3.28), one
has

F
(
∥
√
Kλu(t)∥2

)
≤ E [u0] < F (x1). (3.31)

Direct calculations show that

x1 =
B

A

(
M[φ]

) 1
sc
(
M[u0]

)−αc
,

Now, via (2.13), the inequality (2.17) reads

∥
√

Kλu0∥2 >
B

A
M[φ]

( M[φ]

M[u0]

)αc

= x1.

Thus, the continuity in time with (3.31) gives

∥
√
Kλu(t)∥2 > x1, ∀ t ∈ [0, T ∗).

Then, by (2.13), it follows that

MG[u(t)] > 1, on [0, T ∗). (3.32)

Thus, by the Pohozaev identity BE [φ] = (B − 2)∥
√
Kλφ∥2, it follows that

I[u][M[u]]αc =
(
∥
√

Kλu∥2 −
B

2q
P[u]

)
[M[u]]αc

=
B

2
E [u][M[u]]αc − (

B

2
− 1)∥

√
Kλu∥2[M[u]]αc

≤ B

2
(1− ν)E [φ][M[φ]]αc − (

B

2
− 1)∥

√
Kλφ∥2[M[φ]]αc

≤ −ν(B
2
− 1)∥

√
Kλφ∥2[M[φ]]αc .

The proof of the first point is complete.
(2) Assume that (3.27) fails, then there exists a time sequence {tn} ⊂ [0, T ∗) such that

−εn
(B
2
− 1

)
∥
√
Kλu(tn)∥2 < I[u(tn)] < 0, (3.33)

where εn → 0 and n→ ∞. Moreover, note that

2I[u(tn)] = BE [u(tn)]− (B − 2)∥
√

Kλu(tn)∥2.

Hence, (3.33) implies that

(1− εn)
(
1− 2

B

)
∥
√

Kλu(tn)∥2 < E [u0]. (3.34)

Hence, by (2.13), (2.17), (3.32) and (3.34), we obtain

E [u0]M[u0]
αc > (1− εn)

(
1− 2

B

)
∥
√

Kλu(tn)∥2M[u0]
αc

> (1− εn)
(
1− 2

B

)
∥
√

Kλφ∥2M[φ]αc

> (1− εn)E [φ]M[φ]αc .

(3.35)



12 R. BAI, T. SAANOUNI EJDE-2025/55

Taking n→ ∞ in (3.35), yields

E [u0]M[u0]
αc ≥ E [φ]M[φ]αc . (3.36)

The proof of the second statement (3.27) is achieved by the contradiction of (3.36) with ME [u0] <
1 in (2.17). Hence, this lemma is established. □

3.2. Bi-harmonic case. In this sub-section, one assumes that s = 2.

First case. Assume that (2.16) holds. We start with the next auxiliary result.

Lemma 3.3. (1) The set A− is stable under the flow of (1.1).
(2) There exists ε > 0, such that for any t ∈ [0, T ∗)

I[u(t)] + ε∥∆u(t)∥2 ≤ −B
4

(
m− S[u(t)]

)
. (3.37)

Proof. (1) The proof follows a similar approach to the first point in Lemma 3.1.

(2) Now, taking the scaling uρ := ρ
N
2 u(ρ·) for ρ > 0, we compute

∥uρ∥ = ∥u∥; (3.38)

∥∆uρ∥ = ρ2∥∆u∥; (3.39)

P[uρ] = ρ2BP[u]. (3.40)

Moreover, taking the real function Υ : ρ 7→ S[uρ], we obtain Υ(ρ) = ρ4∥∆u∥2 + ∥u∥2 − ρ2B

p P[u]

and the first derivative reads

Υ′(ρ) = 4ρ3∥∆u∥2 − 2B
ρ2B−1

p
P[u] = 4ρ−1I[uρ]. (3.41)

This implies

ρΥ′(ρ) = 4ρ4∥∆u∥2 − 2B
ρ2B

p
P[u] = 4I[uρ]. (3.42)

Moreover, since B > 2, we obtain(
ρΥ′(ρ)

)′
= 16ρ3∥∆u∥2 − 4B2 ρ

2B−1

p
P[u]

= 2BΥ′(ρ)− 8(B − 2)ρ3∥∆u∥2

≤ 2BΥ′(ρ).

(3.43)

Now, we claim that there exists ρ0 ∈ (0, 1) such that

I[uρ0
] = 0. (3.44)

Indeed, by (3.39) and (3.40), we have

I[uρ] = ρ4
(
∥∆u∥2 − ρ2(B−2)

2p
P[u]

)
:= ρ4Ξ(ρ). (3.45)

Note that Ξ(0) > 0 and Ξ(1) = I[u] < 0. Then there exists ρ0 ∈ (0, 1) such that Ξ(ρ0) = 0,
the claim is proved. Hence, by (3.41), we have Υ′(ρ0) = 0 and Υ(ρ0) = S[uρ0

] ≥ m. Hence, an
integration of (3.43) on [ρ0, 1] gives

Υ′(1)− ρ0Υ
′(ρ0) ≤ 2BΥ(1)− 2BΥ(ρ0).

Note that Υ′(1) = 4I[u], ρ0Υ′(ρ0) = 4I[u(ρ0)] = 0, and Υ(1) = S[u], the above inequality further
implies

I[u] ≤ −B
2

(
m− S[u]

)
. (3.46)

On the other hand, we write

∥∆u∥2 =
B

B − 2

(
S[u]− 2

B
I[u]− ∥u∥2

)
. (3.47)



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 13

Hence, by (3.46), we have that there exists 0 < ε≪ 1, such that

I[u] + ε∥∆u∥2 =
(
1− 2ε

B − 2

)
I[u] + ε

B

B − 2

(
S[u]− ∥u∥2

)
≤ −B

2

(
1− 2ε

B − 2

)(
m− S[u]

)
+ ε

B

B − 2
S[u]

≤ −B
4

(
m− S[u]

)
.

(3.48)

The last statement of Lemma 3.3 is proved by (3.48). □

Now we turn to the proof of the main results. Using the estimate ∥∇γξR∥∞ ≲ R2−|γ|, one has

|
∫
RN

∆2ξR|∇u|2 dx|+ |
∫
RN

∂jk∆ξR∂ju∂kū dx| ≲ R−2∥∇u∥2; (3.49)∣∣ ∫
RN

(∆3ξR)|u|2 dx
∣∣ ≲ R−4. (3.50)

Using estimates (3.49) and (3.50) via Morawetz identity (2.22), one obtains

M ′
R =

4

p

∫
RN

∂kξR∂k

[(
Jα ∗ | · |−τ |u|p

)
|x|−τ

]
|u|p dx+O(R−4) + ∥∇u∥2O(R−2)

− 4N(1− 2

p
)

∫
B(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

− 2
(
(1− 2

p
)

∫
Bc(R)

∆ξR
(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx− 4

∫
RN

∂jkξR∂iku∂ij ū dx
)
.

(3.51)

Denoting the partial derivative ∂
∂xi
u := ui, one obtains via (2.24),∫

RN

∂jkξR∂iku∂ij ū dx =

∫
RN

[(δjk
|x|

− xjxk

|x|3
)
∂rξR +

xjxk

|x|2
∂2r ξR

]
∂iku∂ij ū dx

=

N∑
i=1

∫
RN

|∇ui|2
∂rξR
|x|

dx+

N∑
i=1

∫
RN

|x · ∇ui|2

|x|2
(
∂2r ξR − ∂rξR

|x|

)
dx.

(3.52)

From (3.51) and (3.52), via the equality
∑N

i=1 ∥∇ui∥2 = ∥∆u∥2, it follows that

M ′
R =

4

p

∫
RN

∂kξR∂k

[(
Jα ∗ | · |−τ |u|p

)
|x|−τ

]
|u|p dx+O(R−4) + ∥∇u∥2O(R−2)

+ 16∥∆u∥2 − 4N(1− 2

p
)

∫
B(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

− 2(1− 2

p
)

∫
Bc(R)

∆ξR
(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

+ 8
( N∑

i=1

∫
RN

|∇ui|2
(∂rξR

|x|
− 2

)
dx+

N∑
i=1

∫
RN

|x · ∇ui|2

|x|2
(
∂2r ξR − ∂rξR

|x|
)
dx

)
.

Then, (2.27) gives

M ′
R ≤ 4

p

∫
RN

∂kξR∂k

[(
Jα ∗ | · |−τ |u|p

)
|x|−τ

]
|u|p dx+ cR−2(R−2 + ∥∇u∥2)

+ 16∥∆u∥2 − 4N(1− 2

p
)

∫
B(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

− 2(1− 2

p
)

∫
Bc(R)

∆ξR
(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx.

(3.53)

Take the quantity

(A) :=

∫
RN

∂kξR∂k

[(
Jα ∗ | · |−τ |u|p

)
|x|−τ

]
|u|p dx



14 R. BAI, T. SAANOUNI EJDE-2025/55

= (α−N)

∫
RN

∇ξR
( ·
|x|2

Jα ∗ | · |−τ |u|p
)
|x|−τ |u|p dx

− τ

∫
RN

∇ξR · x
|x|2

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

:= (α−N) · (I)− τ · (II).

In the same way as (3.14), one has

(I) =

∫
B(R)×B(R)

Jα(x− y)|y|−τ |u(y)|p|x|−τ |u(x)|p dx dy

+O
(∫

Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
=

∫
B(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u(x)|p dx+O

(∫
Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
.

From the properties of ξR, one writes

(II) = 2

∫
B(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx+O

(∫
Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
.

Thus,

(A) = 2(−τ − N − α

2
)

∫
B(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u(x)|p dx

+O
(∫

Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
.

Further, (3.53) implies that

M ′
R ≤ 2

(
8

∫
RN

|∆u|2 dx− 2N(1− 2

p
)

∫
RN

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
+

4

p
(A) + cR−2(R−2 + ∥∇u∥2) +O

(∫
Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
= 2

(
8

∫
RN

|∆u|2 dx− 2N(1− 2

p
)

∫
RN

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
+

8

p
(−τ − N − α

2
)

∫
RN

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u(x)|p dx

+O(R−2) +O
(∫

Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
= 16I[u] + cR−2(R−2 + ∥∇u∥2) +O

(∫
Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx

)
.

(3.54)

Since 0 < τ < s
(
1 + α

N

)
, using the Gagliardo-Nirenberg estimate in Proposition 2.2 via the mass

conservation, one writes

∥|x|−τup∥
L

2N
α+N

=
(∫

RN

|x|−
2Nτ
α+N |u|

2Np
α+N dx

)α+N
2N

≲ ∥u∥p−(Np−N−α+2τ
4 )∥∆u∥

Np−N−α+2τ
4

≲ ∥∆u∥
Np−N−α+2τ

4 .

(3.55)



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 15

Now, by the same way as in (3.17), one obtains∫
Bc(R)

(
Jα ∗ | · |−τ |u|p

)
|x|−τ |u|p dx ≲ ∥|x|−τup∥ 2N

α+N
∥|x|−τup∥

L
2N

α+N (Bc(R))

≲ R−τ∥u∥p2Np
α+N

∥|x|−τup∥
L

2N
α+N

≲ R−τ∥∆u∥
Np−N−α

4 ∥∆u∥
Np−N−α+2τ

4

≲ R−τ∥∆u∥
Np−N−α+τ

2 ,

(3.56)

where Np−N−α+τ
2 = B − τ

2 ∈ (0, 2]. Thus, by the interpolation ∥∇u∥2 ≲ ∥∆u∥∥u∥ and Young’s
estimate via (3.54) and (3.56) one obtains

M ′
R ≲ I[u] +R−τ∥∆u∥B− τ

2 +R−2∥∆u∥2 +R−2. (3.57)

Since I[u] < 0, by Gagliardo-Nirenberg estimate in Proposition 2.1 via the mass conservation law,
one has

∥∆u∥2 ≲ P[u] ≲ ∥∆u∥B∥u∥2p−B ≲ ∥∆u∥B .

Thus, B > 2 implies that there exists C1 > 0 such that for any t ∈ [0, T ∗),

∥∆u(t)∥ ≥ C1. (3.58)

Further, (3.37), (3.57) and (3.58), for 2 < B ≤ 2 + τ
2 and R≫ 1, give

M ′
R ≲ I[u] +R−2 +R−2∥∆u∥2 +R−τ∥∆u∥B− τ

2

≲ −∥∆u∥2 +R−2 +R−2∥∆u∥2 +R−τ∥∆u∥B− τ
2

≲ ∥∆u∥2
(
− 1 +R−2 +R−τ∥∆u∥B−2− τ

2

)
≲ −∥∆u∥2.

(3.59)

By time integration, (3.58) and (3.59) imply that

MR[u(t)] ≲ −t, t > T > 0. (3.60)

By time integration again, from (3.59), it follows that

MR[u(t)] ≲ −
∫ t

T

∥∆u(s)∥2 ds, ∀t > T. (3.61)

Now, the definition (2.21) and an interpolation argument give

|MR[u]| = 2|ℑ
∫
RN

ū(∇ξR · ∇u) dx| ≲ R∥∇u∥∥u∥ ≲ R∥∆u∥1/2. (3.62)

So, by (3.60), (3.61) and (3.62), it follows that∫ t

T

∥∆u(s)∥2 ds ≲ |MR[u(t)]| ≲ R∥∆u(t)∥1/2, ∀t > T. (3.63)

Take the real function f(t) :=
∫ t

T
∥∆u(s)∥2. By (3.63), one obtains f4 ≲ f ′. Like previously, this

ODI has no global solution. This completes the proof.

Second case. Assume that (2.17) holds. It is sufficient to prove the next intermediate result.

Lemma 3.4. There exist C > 0 and ε > 0, such that for any t ∈ [0, T ∗), the following statements
hold:

I[u(t)] < −C < 0,

I[u] + ε∥∆u∥2 < 0. (3.64)



16 R. BAI, T. SAANOUNI EJDE-2025/55

Proof. (1) Define the quantity C :=
CN,p,τ,α,λ

p ∥u∥A. Then, by Proposition 2.1, one writes

F (∥∆u(t)∥2) := ∥∆u(t)∥2 − C∥∆u(t)∥B ≤ E [u0], on [0, T ∗). (3.65)

Now, since p > pc gives B > 2, the above real function F has a maximum

F (x1) := F
[( 2

CB

) 2
B−2

]
=

( 2

CB

) 2
B−2

(
1− 2

B

)
.

Moreover, thanks to Pohozaev identities (2.13) and condition (2.17), it follows that

E [u0] <
B − 2

A

(
M[u0]

)−αc
(
M[φ]

)2/sc
. (3.66)

In addition, by (2.12) and the equality sc =
B−2
p−1 , one obtains

F (x1) =
( 2

BC

) 2
B−2

(
1− 2

B

)
=

(
(
A

B
)1−

B
2 (M[φ])p−1(M[u0])

−A/2
) 2

B−2
(
1− 2

B

)
=
B − 2

A

(
(M[u0])

−A/2(M[φ])p−1
) 2

B−2

=
B − 2

A

(
M[u0]

)−αc
(
M[φ]

)2/sc
.

(3.67)

Relations (3.66) and (3.67) imply that E [u0] < F (x1). By the previous inequality and (3.65), one
has

F
(
∥∆u(t)∥2

)
≤ E [u0] < F (x1). (3.68)

Direct calculations show that

x1 =
B

A

(
M[φ]

)2/sc(M[u0]
)−αc

.

Now, the inequality (2.17) reads via (2.13),

∥∆u0∥2 >
B

A
M[φ]

( M[φ]

M[u0]

)αc

= x1.

Thus, the continuity in time with (3.68) give

∥∆u(t)∥2 > x1, ∀ t ∈ [0, T ∗).

Further, one has

MG[u(t)] > 1, for all t ∈ [0, T ∗). (3.69)

Now, by Pohozaev identity (2.13), one has BE [φ] = (B − 2)∥∆φ∥2. So, it follows that for some
0 < ν < 1,

I[u][M[u]]αc =
(
∥∆u∥2 − B

2p
P[u]

)
[M[u]]αc

=
B

2
E [u][M[u]]αc − (

B

2
− 1)∥∆u∥2[M[u]]αc

≤ B

2
(1− ν)E [φ][M[φ]]αc − (

B

2
− 1)∥∆φ∥2[M[φ]]αc

≤ −ν(B
2
− 1)∥∆φ∥2[M[φ]]αc .

The proof of the first point is complete.
(2) Assume that (3.64) fails, then there exists a time sequence {tn} ⊂ [0, T ∗), such that

−εn
(B
2
− 1

)
∥∆u(tn)∥2 < I[u(tn)] < 0, (3.70)

where εn → 0 as n→ ∞. Moreover, note that

2I[u(tn)] = BE [u(tn)]− (B − 2)∥∆u(tn)∥2.



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 17

Hence, (3.70) implies that

(1− εn)
(
1− 2

B

)
∥∆u(tn)∥2 < E [u0]. (3.71)

Further, by (2.13), (2.17), (3.69) and (3.71), we obtain

E [u0]M[u0]
αc > (1− εn)

(
1− 2

B

)
∥∆u(tn)∥2M[u0]

αc

> (1− εn)
(
1− 2

B

)
∥∆φ∥2M[φ]αc

> (1− εn)E [φ]M[φ]αc .

(3.72)

Taking n→ ∞ in (3.72), we obtain

E [u0]M[u0]
αc ≥ E [φ]M[φ]αc . (3.73)

The proof of (3.37) is achieved by the contradiction of (3.73) with (2.17). Hence, this lemma is
established. □

4. Schrödinger equation with local source term

In this section, we establish Theorem 2.4.

4.1. Schrödinger equation with inverse square potential. In this subsection, we take s = 1.

First case. One keeps previous notation and assume that (2.18) holds. We start with the next
auxiliary result which can be proved arguing as in Lemma 3.1.

Lemma 4.1. (1) The set A′− is stable under the flow of (1.2).
(2) There exists ε > 0, such that for any t ∈ [0, T ∗),

J [u(t)] + ε∥
√

Kλu(t)∥2 ≤ −B
′

4

(
m′ − S ′[u(t)]

)
. (4.1)

Proposition 2.6 via (2.24) and (2.25) gives

M ′
R[u] = 4

∫
RN

[(δlk
r

− xlxk
r3

)
∂rξR +

xlxk
r2

∂2r ξR

]
ℜ(∂ku∂lū) dx−

∫
RN

∆2ξR|u|2 dx

+ 4λ

∫
RN

∂rξR
|u|2

|x|3
dx+ 2(

1

q
− 1)

∫
RN

(∂2r ξR +
N − 1

r
∂rξR)|x|−2τ |u|2q dx

− 4τ

q

∫
RN

∂rξR
r

|x|−2τ |u|2q dx

= 4

∫
RN

[( |∇u|2
r

− |x · ∇u|2

r3

)
∂rξR +

|x · ∇u|2

r2
∂2r ξR

]
dx−

∫
RN

∆2ξR|u|2 dx

+ 4λ

∫
RN

∂rξR
|u|2

|x|3
dx+ 2(

1

q
− 1)

∫
RN

(
∂2r ξR + (N − 1 +

2τ

q − 1
)
∂rξR
r

)
|x|−2τ |u|2q dx.

Now, noting that λ ≥ 0, by (1.5) and (2.27), one obtains

M ′
R[u]− 8J [u] = 4

∫
RN

[( |∇u|2
r

− |x · ∇u|2

r3

)
∂rξR +

|x · ∇u|2

r2
∂2r ξR

]
dx

−
∫
RN

∆2ξR|u|2 dx+ 4λ

∫
RN

∂rξR
|u|2

|x|3
dx− 8∥

√
Kλu∥2

− 2
q − 1

q

∫
RN

(
∂2r ξR + (N − 1 +

2τ

q − 1
)
∂rξR
r

− 2
B′

q − 1

)
|x|−2τ |u|2q dx

≤ 4

∫
RN

|x · ∇u|2

r2

(
∂2r ξR − ∂rξR

r

)
dx−

∫
RN

∆2ξR|u|2 dx

− 2
q − 1

q

∫
RN

(
∂2r ξR + (N − 1 +

2τ

q − 1
)
∂rξR
r

− 2
B′

q − 1

)
|x|−2τ |u|2q dx.



18 R. BAI, T. SAANOUNI EJDE-2025/55

Using the estimate ∥∇γξR∥∞ ≲ R2−|γ| via the mass conservation law, one has∣∣ ∫
RN

∆2ξR|u|2 dx
∣∣ ≲ R−2. (4.2)

Moreover, one decomposes the above quantity as follows

M ′
R[u]− 8J [u] ≤ 4

∫
RN

|x · ∇u|2

r2

(
∂2r ξR − ∂rξR

r

)
dx−

∫
RN

∆2ξR|u|2 dx

− 2
q − 1

q

∫
RN

(
∂2r ξR + (N − 1 +

2τ

q − 1
)
∂rξR
r

− 2
B′

q − 1

)
|x|−2τ |u|2q dx

:= −(A1)− 2
q − 1

q
· (A2).

(4.3)

By the properties of ξR,

∂2r ξR + (N − 1 +
2τ

q − 1
)
∂rξR
r

− 2
B′

q − 1
= 0, for B(R).

Thus, by the Gagliardo-Nirenberg estimate via the mass conservation law, (2.27) and (1.5), one
obtains

(A2) =

∫
Bc(R)

(
∂2r ξR + (N − 1 +

2τ

q − 1
)
∂rξR
r

− 2
B′

q − 1

)
|x|−2τ |u|2q dx

≲ R−2τ

∫
RN

|u|2q dx

≲ R−2τ∥
√
Kλu∥B

′−2τ∥u∥2q−B′+2τ

≲ R−2τ∥
√
Kλu∥B

′−2τ .

(4.4)

Since J [u] < 0, by the Gagliardo-Nirenberg estimate in Proposition 2.2 via the mass conservation
law, one has

∥
√
Kλu∥2 ≲

∫
RN

|x|−2τ |u|2q dx ≲ ∥
√

Kλu∥B
′
∥u∥2q−B′

≲ ∥
√
Kλu∥B

′
.

Thus, B′ > 2 implies that there is C2 > 0 such that for t ∈ [0, T ∗),

∥
√

Kλu(t)∥ ≥ C2. (4.5)

Thus, (4.3)-(4.4) and (4.1) give for 2 < B′ ≤ 2 + 2τ and R≫ 1,

M ′
R[u] ≲ J [u] +R−2 +R−2τ∥

√
Kλu∥B

′−2τ

≲ −∥
√
Kλu∥2 +R−2 +R−2τ∥

√
Kλu∥B

′−2τ

≲ ∥
√
Kλu∥2

(
− 1 +R−2 +R−2τ∥

√
Kλu∥B

′−2−2τ
)

≲ −∥
√
Kλu∥2.

(4.6)

By time integration, (4.5), and (4.6) imply that

MR[u(t)] ≲ −t, t > T > 0. (4.7)

By time integration again, from (4.6) and (4.7), it follows that

MR[u(t)] ≲ −
∫ t

T

∥
√

Kλu(s)∥2 ds. (4.8)

Now, the definition (2.21) via the mass conservation law gives

|MR[u]| = 2|ℑ
∫
RN

ū(∇ξR · ∇u) dx| ≲ R∥∇u∥∥u∥ ≲ R∥∇u∥. (4.9)

By (4.7), (4.8) and (4.9), it follows that∫ t

T

∥
√
Kλu(s)∥2 ds ≲ |MR[u(t)]| ≲ R∥

√
Kλu∥, ∀t > T. (4.10)



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 19

Take the real function f(t) :=
∫ t

T
∥
√
Kλu(s)∥2. By (4.10), one obtains f2 ≲ f ′. Like previously,

this ODI has no global solution. This completes the proof.

Second case. The proof is similar to the previous section.

4.2. Bi-harmonic case. In this sub-section, one assumes that s = 2.

First case. Assume that (2.18) holds. We start with the next auxiliary result which can be proved
arguing as in Lemma 3.3.

Lemma 4.2. (1) The set A′− is stable under the flow of (1.2).
(2) There exists ε > 0, such that for any t ∈ [0, T ∗),

J [u(t)] + ε∥∆u(t)∥2 ≤ −B
′

4

(
m′ − S ′[u(t)]

)
. (4.11)

Using the estimates (3.49) and (3.50) via Morawetz identity (2.23), one obtains

−M ′
R = 2

∫
RN

(
2∂jk∆ξR∂ju∂kū− 1

2
(∆3ξR)|u|2 − 4∂jkξR∂iku∂ij ū

+∆2ξR|∇u|2 +
q − 1

q
(∆ξR)|x|−2τ |u|2q − 1

q
∇ξR · ∇(|x|−2τ )|u|2q

)
dx

=
8B′

q

∫
B(R)

|x|−2τ |u|2q dx− 8

∫
RN

∂jkξR∂iku∂ij ū dx+O(R−4) + ∥∇u∥2O(R−2)

+ 2
q − 1

q

∫
Bc(R)

(∆ξR)|x|−2τ |u|2q dx− 2

q

∫
Bc(R)

∇ξR · ∇(|x|−2τ )|u|2q dx.

(4.12)

Thus, by (3.52), one writes

−M ′
R =

8B′

q

∫
B(R)

|x|−2τ |u|2q dx− 2

q

∫
Bc(R)

∇ξR · ∇(|x|−2τ )|u|2q dx

+ 2
q − 1

q

∫
Bc(R)

(∆ξR)|x|−2τ |u|2q dx+O(R−4) + ∥∇u∥2O(R−2)

− 8

N∑
i=1

∫
RN

|∇ui|2
∂rξR
|x|

dx− 8

N∑
i=1

∫
RN

|x · ∇ui|2

|x|2
(
∂2r ξR − ∂rξR

|x|

)
dx

= −16J [u]− 8B′

q

∫
Bc(R)

|x|−2τ |u|2q dx+O(R−4) + ∥∇u∥2O(R−2)

− 2

q

∫
Bc(R)

∇ξR · ∇(|x|−2τ )|u|2q dx+ 2
q − 1

q

∫
Bc(R)

(∆ξR)|x|−2τ |u|2q dx

− 8
( N∑

i=1

∫
RN

|∇ui|2
(∂rξR

|x|
− 2

)
dx+

N∑
i=1

∫
RN

|x · ∇ui|2

|x|2
(
∂2r ξR − ∂rξR

|x|
)
dx

)
.

Then, by an interpolation argument and Young estimate, (2.27) gives

M ′
R ≲ J [u] +O

(∫
Bc(R)

|x|−2τ |u|2q dx
)
+R−2 +R−2∥∆u∥2. (4.13)

Since 1 < q < N
N−4 , by the Gagliardo-Nirenberg inequality, one writes∫

Bc(R)

|x|−2τ |u|2q dx ≤ cR−2τ∥u∥2q2q ≤ cR−2τ∥∆u∥N
2 (q−1).

So,

M ′
R[u] ≲ J [u] +R−2τ∥∆u∥B

′−τ +R−2. (4.14)

Since J [u] < 0, by the Gagliardo-Nirenberg estimate in Proposition 2.2 via the mass conservation
law, one has

∥∆u∥2 ≲ Q[u] ≲ ∥∆u∥B
′
∥u∥2q−B′

≲ ∥∆u∥B
′
.



20 R. BAI, T. SAANOUNI EJDE-2025/55

Thus, B′ > 2 implies that there is C3 > 0 such that for any t ∈ [0, T ∗),

∥∆u(t)∥ ≥ C3. (4.15)

Thus, (4.11)-(4.15) give for 2 < B′ ≤ 2 + τ and R≫ 1,

M ′
R[u] ≲ J [u] +R−2 +R−2∥∆u∥2 +R−2τ∥∆u∥B

′−τ

≲ −∥∆u∥2 +R−2 +R−2∥∆u∥2 +R−2τ∥∆u∥B
′−τ

≲ ∥∆u∥2
(
− 1 +R−2 +R−2τ∥∆u∥B

′−2−τ
)

≲ −∥∆u∥2.

(4.16)

By time integration, (4.15) and (4.16) imply that

MR[u(t)] ≲ −t, t > T > 0 (4.17)

By time integration again, from (4.16) and (4.17), it follows that

MR[u(t)] ≲ −
∫ t

T

∥∆u(s)∥2 ds. (4.18)

The rest of the proof follows as previously.

Acknowledgements. R. Bai was supported by the Postdoctoral Fellowship Program of CPSF
(Grant No. GZC20230694). The authors want to thank the anonymous referees for the helpful
comments and suggestions.

References

[1] M. G. Alharbi. T. Saanouni; Sharp threshold of global well-posedness vs finite time blow-up for a class of

inhomogeneous Choquard equations, J. Math. Phys., 60 (8), 2019, 24.

[2] J. An, R. Jang, J. Kim; Global existence and blow-up for the focusing inhomogeneous nonlinear schrödinger
equation with inverse-square potential, Discr. Cont. Dyn. Syst.- Ser. B., 28 (2), 2023, 1046-1067.

[3] J. An, J. Kim, P. Ryu; Local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation

in Sobolev spaces, Z. Anal. Anwend., 41 (1/2), 2022, 239-258.
[4] R. Bai, B. Li; Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equa-

tion, Nonlinear Anal., 232 (2023), 113266.

[5] A. A. Balinsky, W. D. Evans; Some Recent Results on Hardy-Type Inequalities, Appl. Math. Inf. Sci., 4 (2),
2010, 191-208.

[6] J. Belmonte-Beitia, V. M. Pérez-Garćıa, V. Vekslerchik, P. J. Torres; Lie symmetries and solitons in nonlinear
systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett., 98 (6), 2007, 064102.

[7] L. Bergé; Soliton stability versus collapse, Phys. Rev. E., 62 (3), 2000, 3071-3074.

[8] T. Boulenger, E. Lenzmann; Blowup for biharmonic NLS, Ann. Sci. Éc. Norm. Supér., 50 (3), 2017, 503–544.

[9] L. Campos, C. M. Guzmán; On the inhomogeneous NLS with inverse-square potential, Z. Angew. Math. Phys.,

72 (4), 2021, 29.
[10] L. Campos, C.M. Guzmán; Scattering for the non-radial inhomogenous biharmonic NLS equation. Calc. Var.,

61 (4), 2022, 15.

[11] M. Cardoso, L. G. Farah; Blow-up of non-radial solutions for the L2 critical inhomogeneous NLS equation,
Nonlinearity, 35 (8), 2022.

[12] Y. Cho, T. Ozawa, C. Wang; Finite time blowup for the fourth-order NLS, Bull. Korean Math. Soc., 53 (2),

2016, 615-640.
[13] V.D. Dinh; Non-radial finite time blow-up for the fourth-order nonlinear Schrödinger equations, Appl. Math.

Lett., 132, 2022.
[14] V. D. Dinh, S. Keraani; Energy scattering for a class of inhomogeneous biharmonic nonlinear Schrödinger

equations in low dimensions, arXiv:2211.11824, 2022.

[15] D. Du, Y. Wu, K. Zhang; On blow-up criterion for the nonlinear Schrödinger equation, Discrete Contin. Dyn.
Sys., 36 (7), 2016, 3639-3650.

[16] R. T. Glassey; On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J.

Math. Phys., 18 (9), 1977, 1794-1797.
[17] C. M. Guzmán, A. Pastor; On the inhomogeneous biharmonic nonlinear Schrödinger equation: local, global

and stability results, Nonlinear Anal. Real World Appl., 56, 2020.

[18] C. M. Guzmán, A. Pastor; Some remarks on the inhomogeneous biharmonic NLS equation, Nonlinear Anal.
Real World Appl., 67, 2022.

[19] J. Holmer, S. Roudenko; Divergence of infinite-variance nonradial solutions to 3D NLS equation, Comm.

Partial Differ. Equa., 35 (5), 2010, 878-905.



EJDE-2025/55 INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATION 21

[20] R. Jang, J. An, J. Kim; The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger

equation with inverse-square potential, arXiv:2107.09826, 2021.

[21] H. Kalf, U.-W. Schmincke, J. Walter, R. Wüst; On the spectral theory of Schrödinger and Dirac operators
with strongly singular potentials, Spectral Theory and Differential Equations (Proceedings Symposium Dundee,

1974; dedicated to Konrad Jörgens), Lecture Notes in Mathematics, Vol. 448, Springer, Berlin, (1975), 182-226.

[22] V. I. Karpman; Stabilization of soliton instabilities by higher-order dispersion: fourth-order nonlinear
Schrödinger- type equations, Physical Review E., 53 (2), 1996, R1336.

[23] V. I. Karpman, A. G. Shagalov; Stability of solitons described by nonlinear schrödinger-type equations with

higher-order dispersion, Physica D: Nonlinear Phenomena, 144 (1-2), 2000, 194-210.
[24] Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, M. R. Belic, L. Torner; Rotating vortex clusters in media

with inhomogeneous defocusing nonlinearity, Opt. Lett., 42 (3), 2017, 446–449.

[25] S. Kim; On well-posedness for inhomogeneous Hartree equations in the critical case,Comm. Pur. Appl. Anal.,
22 (7), (2023), 2132-2145.

[26] S. Kim, Y. Lee, I. Seo; Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations,
Nonlinear Anal. 240 (2024), 113463.

[27] E. Lieb, M. Loss; Analysis, 2nd ed. Graduate Studies in Mathematics, Vol. 14, American Mathematical Society,

Providence, RI, 2001.
[28] Y. Martel; Blow-up for the nonlinear Schrödinger equation in nonisotropic spaces, Nonlinear Anal. 28 (12),

1997, 1903-1908.

[29] F. Merle; Nonexistence of minimal blow-up solutions of equations iut = −∆u−k(x)|u|4/Nu in RN , Ann. Inst.
H. Poincaré Phys. Théor., 64 (1), 1996, 33-85.

[30] T. Ogawa, Y. Tsutsumi; Blow-up of H1 solution for the nonlinear Schrödinger equation, J. Differ. Equ., 92

(2), 1991, 317-330.
[31] T. Ogawa, Y. Tsutsumi; Blow-up of H1 solutions for the one-dimensional nonlinear Schrödinger equation

with critical power nonlinearity, Proc. Am. Math. Soc., 111 (2), 1991, 487-496.

[32] L. E. Payne, D. H. Sattinger; Saddle points and instability of non-linear hyperbolic equations, Isr. J. Math.,
22 (1976), 273-303. DOI: 10.1007/BF02761595

[33] F. Planchon, J. G. Stalker, A. S. Tahvildar-Zadeh; Lp estimates for the wave equation with the inverse-square

potential, Discrete Contin. Dyn. Syst., 9 (2), 2003, 427-442.
[34] T. Saanouni; Energy scattering for radial focusing inhomogeneous bi-harmonic Schrödinger equations, Calc.

Var., 60 (3), 2021.
[35] T. Saanouni, A. Boubaker; Inhomogeneous generalized Hartree equation with inverse square potential, SeMA

81, 2024, 679-706.

[36] T. Saanouni, R. Ghanmi; A note on the inhomogeneous fourth-order Schrödinger equation, J. Pseudo-Differ.
Oper. Appl., 13 (4), 2022.

[37] T. Saanouni, R. Ghanmi; Local well-posedness of a critical inhomogeneous bi-harmonic Schrödinger equation,

Advances in Opertor Theory., 9 (2024), 2662-2009.
[38] T. Saanouni, C. Peng; Local well-posedness of a critical inhomogeneous bi-harmonic Schrödinger equation,

Mediterr. J. Math., 20 (2023), 170.

[39] T. Saanouni, C. Xu; Scattering Theory for a Class of Radial Focusing Inhomogeneous Hartree Equations.
Potential Anal., 58, 2023, 617-643.

[40] T. Suzuki; Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory,

2 (3), 2013, 531-542.
[41] T. Suzuki; Solvability of nonlinear Schrödinger equations with some critical singular potentialvia generalized

Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (1), 2016, 1-34.
[42] E. C. Titchmarsh; Eigen function Expansions Associated with Second-Order Differential Equations, University

Press, Oxford, 1946.
[43] C. Xu; Scattering for the non-radial focusing inhomogeneous nonlinear Schrödinger-Choquard equation,

arXiv:2104.09756, 2021.

Ruobing Bai

School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

Email address: baimaths@hotmail.com

Tarek Saanouni

Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia
Email address: t.saanouni@qu.edu.sa


	1. Introduction
	2. Background and main results
	2.1. Preliminaries
	2.2. Main results
	2.3. Useful estimates

	3. Schrödinger equation with non-local source term
	3.1. Schrödinger equation with inverse square potential
	3.2. Bi-harmonic case

	4. Schrödinger equation with local source term
	4.1. Schrödinger equation with inverse square potential
	4.2. Bi-harmonic case
	Acknowledgements

	References