Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 53, pp. 1–15.

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

DOI: 10.58997/ejde.2025.53

SOLUTIONS WITH EXPANDING COMPACT SUPPORT OF SATURATED

SCHRÖDINGER EQUATIONS: SELF-SIMILAR SOLUTIONS

PASCAL BÉGOUT, JESÚS ILDEFONSO DÍAZ

Abstract. We prove the existence of solutions u(t, x) of the Schrödinger equation with a sat-

uration nonlinear term (u/|u|) having compact support, for each t > 0, that expands with a

growth law of the type C
√
t. The primary tool is considering the self-similar solution of the

associated equation.

Contents

1. Introduction 1
2. Self-similar solutions 3
3. Existence and uniqueness of the solutions 7
4. Setting of the framework and proofs of the existence theorems 8
4.1. Homogeneous Dirichlet boundary condition with a domain of finite measure 9
4.2. General case 10
4.3. Proofs of the existence and compactness theorems 11
5. Appendix 13
Acknowledgements 14
References 14

1. Introduction

The existence of compactly support solutions to Schrödinger equation was a constant subject
of research since Schrödinger postulated the existence of such equation in 1925 and published it in
1926. For the case of the linear equation it seems that it was Sir Nevill Francis Mott (1905-1996),
who would later win the Nobel Prize in 1977, proposed the study of the infinite well potential in
his 1930 book [17]. This was a generalization of the finite well potential proposed, in 1928, by
George Gamow [15] when finding the tunnel effect by first time in the literature. Solutions of the
linear Schrödinger equation with an infinite well potential have compact support (the compact
set of RN where the potential is finite) but the mathematical study of this problem presents
some ambiguities [13] which disappear when such a discontinuous potential is replaced by strongly
singular potentials of the Pöschl-Teller type [13, 14, 18].

This study of the support of solutions of nonlinear Schrödinger equations was also considered
by many authors but with negative results when the nonlinear term is Lipschitz continuous (see,
e.g., the presentation made by Bourgain in [9]). These authors made completely new contributions

2020 Mathematics Subject Classification. 35C06, 35A01, 35A02, 35J91, 35Q55.
Key words and phrases. Schrödinger equation with saturated nonlinearity; solutions compactly supported;

energy method; Dirichlet boundary condition; Neumann boundary condition; existence; uniqueness.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted April 14, 2025. Published May 24, 2025.

1



2 P. BÉGOUT, J. I. DÍAZ EJDE-2025/53

in the subject by showing that solutions with compact support do exist when the nonlinear term
is not Lipschitz continuous but of the form

i
∂u

∂t
+∆u = a|u|−(1−m)u+ f(t, x), (1.1)

for some m ∈ (0, 1) and for a suitable complex coefficient a. This equation is associated to the
consideration of the non-Kerr law optical Schrödinger equation arising, for instance, in nonlinear
optical media. This type of equation also arises in QuantumMechanics and Hydrodynamics. When
searching for “solitary wave solutions” of the form u(t, x) = ψ(x)eibt (when f(t, x) = eibtF (x))
then the complex function u satisfies a stationary nonlinear equation which leads to solutions with
compact support once we assume that F (x) has compact support.

The above problem was extended to the case of saturated nonlinear terms (m = 0) in the recent
paper [7] proving that “solitary wave solutions” u(t, x) = ψ(x)eibt have compact support even if
F (x) does not have a compact support but is small enough outside of some compact subset of
RN . From the qualitative point of view, the above type of solutions with compact support (for
the mentioned linear and nonlinear cases) concern some special type of solutions: “solitary wave
solutions” of the form u(t, x) = ψ(x)eibt which implies that support of u(t) does not move, for any
t > 0, since suppu(t) = suppψ.

A different point of view was followed by the authors in [4] where the existence of a self-
similar solution of the form u(t, x) = tp/2φ

(
x/

√
t
)
was proved for equations of the type (1.1) with

m ∈ (0, 1) once we assume that f(t, x) = t
p−2
2 F

(
x/

√
t
)
: it was proved in that paper that if suppF

is compact then the solution profile φ is also compact. As it was detailed later, for this type of
solution their support suppu(t) expands with time t > 0, with a sublinear growth of the type
C
√
t.
The main objective of this article is to extend the results of [4] to the saturated case (m = 0)

by showing that the corresponding solution has an expanding support suppu(t) that expands
with time t > 0, with a sublinear growth of type C

√
t even if the profile F of the data f(t, x) =

t
p−2
2 F

(
x/

√
t
)
is not compactly supported but, as in [7], is sufficiently small outside a compact

subset of RN . One of the consequences of such general assumption on f(t, x) is that we can
extend the property of solutions with compact support when we couple the Schrödinger equation
with some other phenomena (as for instance the existence of some magnetic fields: see Section 9
of [7]). Here we are interested in finding self-similar solutions with compact support in the space
variable of the following Schrödinger equation with saturated nonlinearity,

i
∂u

∂t
+∆u = aU + f(t, x), (t, x) ∈ (0,∞)× RN ,

U =
u

|u|
, a.e. in

{
(t, x) ∈ (0,∞)× RN ;u(t, x) ̸= 0

}
,

(1.2)

where a ∈ C. For this, it is enough to study the equation satisfied by the profile φ of u, that is

−∆φ+ aΦ− ip

2
φ+

i

2
x.∇φ = −F, in D ′(RN ),

Φ =
φ

|φ|
, a.e. in

{
x ∈ RN ;φ(x) ̸= 0

}
,

(1.3)

where p ∈ C with Re(p) = 2, φ = u(1), and F = f(1). We will maintain the notation and several
common arguments with our previous papers [4] and [7], but new results will be given, improving
both papers. As in [4], it is useful to introduce a change of unknowns which brings us back to the
search for solutions to the problem

−∆g + aG− i
N + 2p

4
g − 1

16
|x|2g = −Fe−i

|x|2
8 , in D ′(RN ),

G =
g

|g|
, a.e. in

{
x ∈ RN ; g(x) ̸= 0

}
.

(1.4)



EJDE-2025/53 SATURATED SCHRÖDINGER EQUATIONS 3

So in this paper, we study the following which is more general equation than (1.4),

−∆u+ aU + b u+ V u = F, in H−1(Ω) + L∞(Ω),

U =
u

|u|
, a.e. in ω =

{
x ∈ Ω;u(x) ̸= 0

}
,

(1.5)

where (a, b) ∈ C2, and Ω is a subset of RN whose boundary is Γ, with homogeneous Dirichlet
boundary condition

u|Γ = 0, (1.6)

or with homogeneous Neumann boundary condition

∂u

∂ν |Γ
= 0. (1.7)

The compactness of the support of solutions will be obtained by some improvements of the energy
methods presented in the monograph [2] (see also the extension to some variational inequalities
made in [12]). We mention that the method such as it was developed in the above mentioned
references is only well adapted to its application to complex problems of Ginzburg -Landau type
[1] in which the time derivative of the unknown contains a real part (situation which is not valid
for the Schrödinger equation).

The organization of this article is as it follows: Section 2 is devoted to the structure of self-
similar solutions and to the presentation of the main result of this paper (Theorem 2.3 below).
Details on the notion of solutions, the results on the existence and uniqueness of solutions are
collected in Section 3. A set of auxiliary results preparing the application of an energy method
leading to the compactness of the support of the solution, as well as the proof of the results stated
in the previous sections are presented in Section 4. Finally, an Appendix is devoted to the proof
of the additional regularity obtained from the structure of self-similar solutions.

As indicated before, this paper extends some previous papers by the authors ([4] and a part of
[6]) to the case m = 0. Nevertheless, since the applied techniques are of a different type, they do
not allow to conclude some previous results in their complete generality. Furthermore, despite the
fact that [7] also concerns equation (1.5), we point out that the assumptions and results differ so
that they cannot be employed to construct self-similar solutions with compact support in space.
For instance, Theorem 3.2 vs [7, Theorem 2.6]: [7, Theorem 2.6] in less restrictive in terms of
V , and Theorem 3.2, only considers the Dirichlet condition and |Ω| < ∞. But Theorem 3.2 is
more general in terms of (a, b) since (a, b) ∈ C × B while in [7, Theorem 2.6], (a, b) ∈ A2 satisfy
some additional conditions, and A ⊊ B. Theorem 3.4 vs [7, Theorem 2.6]: Theorem 3.4 is more
restrictive in terms of (a, b) but it allows V to be a complex-valued function with no sign restriction
about Re(V ), while in [7, Theorem 2.6], V is a nonnegative real-valued functions. It is essential
to allow for choosing V with a negative real part to consider self-similar solutions.

Here is a list of symbols we will use in this paper: for a complex number z, we denote by z,
Re(z) and Im(z), its conjugate, real and imaginary part, respectively, and i2 = −1. N0 = N∪{0}.
For p ∈ [1,∞], p′ is the conjugate of p defined by 1

p + 1
p′ = 1. Unless specified, all functions are

complex-valued and all the vector spaces are considered over the field R. For a Banach space X,
we denote by X⋆ := L (X;R) its topological dual and by ⟨·, ·⟩X⋆,X the X⋆ −X duality product.
By convention, W 0,q(RN ) = Lq(RN ), for any 0 < q < ∞. For positive parameters a1, . . . , an,
we shall write C(a1, . . . , an) to indicate that C is a positive constant which depends only and
continuously on a1, . . . , an. Finally, if A is a subset of RN then Ac denotes its complement, and
A \B = A ∩Bc.

Let us recall that if X and Y are two Banach spaces 1 such that X ↪→ Y with dense em-
bedding then Y ⋆ ↪→ X⋆, and for any F ∈ Y ⋆ and u ∈ X, ⟨F, u⟩X⋆,X = ⟨F, u⟩Y ⋆,Y . By

the Riesz representation Theorem, we have for any p ∈ [1,∞), F ∈ Lp′
(Ω) and u ∈ Lp(Ω),

⟨F, u⟩Lp′ (Ω),Lp(Ω) = Re
∫
Ω
F (x)u(x)dx. In particular, this implies that we shall always identify

L2(Ω) with its topological dual. In addition, if A1 and A2 are two Banach spaces such that
A1, A2 ⊂ H for some Hausdorff topological vector space H, and if A1 ∩ A2 is dense in both A1

1Actually, locally convex topological vector spaces is enough which allows to consider X = D(Ω).



4 P. BÉGOUT, J. I. DÍAZ EJDE-2025/53

and A2 then A1 ∩ A2 and A1 + A2 are Banach spaces, and
(
A1 ∩ A2

)⋆
= A⋆

1 + A⋆
2. This justifies

the identity (3.1) below. For more details, see Trèves [19], Bergh and Löfström [8, 3].

2. Self-similar solutions

Let us recall that the notion of self-similar solutions relies on the transformation λ 7→ (uλ, U
λ),

where for λ > 0, p ∈ C, u ∈ L1
loc

(
(0,∞) × RN

)
and U a saturated section associated to u

(Definition 2.1 below),

uλ(t, x) = λ−pu(λ2t, λx), (2.1)

Uλ(t, x) = λ−(p−2)U(λ2t, λx), (2.2)

for a.e. (t, x) ∈ (0,∞) × RN . We also recall that λp := ep lnλ and |λp| = λRe(p). If Re(p) = 2
then a straightforward calculation shows that if (u, U) is a solution to (1.2) with f = 0, then so
is (uλ, U

λ), for any λ > 0. In particular, Uλ is a saturated section associated to uλ. To keep this
property when f ̸= 0, with f ∈ L1

loc

(
(0,∞)× RN

)
, we assume that f satisfies

∀λ > 0, fλ = f, (2.3)

or equivalently,

f(t, x) = t
p−2
2 F

(
x√
t

)
, (2.4)

for a.e. (t, x) ∈ (0,∞)× RN , where F = f(1). To have functions f satisfying (2.3), it is sufficient
for any given function F ∈ L1

loc(RN ) to define f by (2.4). Furthermore, we easily check that (u, U)
satisfies the invariance property

∀λ > 0, (uλ, U
λ) = (u, U),

if, and only if,

u(t, x) = tp/2φ

(
x√
t

)
, (2.5)

U(t, x) = t
p−2
2 Φ

(
x√
t

)
, (2.6)

for a.e. (t, x) ∈ (0,∞) × RN , where (φ,Φ) = (u(1), U(1)). This remarkable invariance property
leads to the well-known definition of self-similar solution.

Definition 2.1. Let θ ⊆ RN be an open subset and let u ∈ L1
loc(θ). A function U ∈ L∞(θ) is

said to be a saturated section associated to u if ∥U∥L∞(θ) ≤ 1 and U = u/|u|, almost everywhere

in ω :=
{
y ∈ θ;u(y) ̸= 0

}
.

Definition 2.2. Let f ∈ C
(
(0,∞);L2(RN )

)
satisfy (2.3) and let p ∈ C be such that Re(p) = 2. A

solution (u, U) to (1.2) is said to be self-similar if u ∈ C
(
(0,∞);L2(RN )

)
, U is a saturated section

associated to u and if for any λ > 0, (uλ, U
λ) = (u, U), where uλ and Uλ are defined by (2.1) and

(2.2), respectively. In this cases, u(1) is called the profile of u and is denoted by φ.

It follows from (1.2), (2.5) and (2.6) that the profile φ of u and Φ satisfy (1.3). In particular,
Φ is a saturated section associated to φ. Conversely, if (φ,Φ) ∈ L2(RN )× L∞(RN ) satisfies (1.3)
with ∥Φ∥L∞(RN ) ≤ 1, then the functions u and U defined by (2.5) and (2.6), respectively, belong

to C
(
(0,∞);L2(RN )

)
(Lemma 5.1) and L∞((0,∞) × RN ), respectively, U is a saturated section

associated to u and u is a self-similar solution to (1.2), where f is defined by (2.4) and satisfies
(2.3). A priori estimates on φ are not easy to obtain due to the term x.∇φ. Thus, in the literature,
this problem is circumvented using the bijective transformation

g(x) = φ(x)e−i
|x|2
8 , for a.e. x ∈ RN . (2.7)

The saturated section Φ associated to φ then becomes

G(x) = Φ(x)e−i
|x|2
8 , for a.e. x ∈ RN . (2.8)



EJDE-2025/53 SATURATED SCHRÖDINGER EQUATIONS 5

It follows that for any p ∈ C and φ ∈ L2(RN ), whose saturated section associated to φ is Φ, (φ,Φ)
is a solution to (1.3) if, and only if, (g,G) ∈ L2(RN ) × L∞(RN ) is a solution to (1.4) and G is a
saturated section associated to g. The study of (1.4) is then more convenient than that of (1.3),
and is related to Theorem 3.4. Let

A = C \
{
z ∈ C; Re(z) ≤ 0 and Im(z) = 0

}
. (2.9)

The main result of this paper is the following.

Theorem 2.3. Assume that a ∈ A is such that Im(a) ≤ 0. Let p ∈ C be such that Re(p) = 2,
let f ∈ C

(
(0,∞);L2(RN )

)
satisfy (2.3) and set F = f(1). Assume also that F|Kc ∈ L∞(Kc), for

some compact subset K of RN .

(1) Existence. For any R > 0 such that K ⊂ B(0, R) and any ε > 0, there exist M =
M(|a|, | Im(p)|, R,N) and δ = δ(|a|, | Im(p)|, R, ε,N) satisfying the following property.
If ∥F∥L2(RN ) ≤ δ and ∥F∥L∞(Kc) ≤ 1

M , then there exists a self-similar solution (u, U)
to (1.2) such that

u ∈ C
(
(0,∞);H2(RN )

)
∩ C1

(
(0,∞);H1(RN )

)
∩ C2

(
(0,∞);L2(RN )

)
(2.10)

and for any t > 0, suppu(t) is compact. In addition, the profile φ of u satisfies that
suppφ ⊂ K(ε) ⊂ B(0, R+ ε), where

K(ε) =
{
x ∈ RN ; dist(x,K) ≤ ε

}
,

which is compact.
(2) Uniqueness. Let (u, U) and (v, V ) be two self-similar solutions to (1.2) with profiles φ

and ϕ, respectively, and with suppφ∪ suppϕ ⊂ B(0, r), for some r > 0. Assume that one
of the two following conditions is satisfied.
(a) Re(a) = 0.

(b) Re(a) > 0 and r2 ≤ 8 Im(p) + 4 | Im(a)|
Re(a) (N + 4).

Then for any t > 0, u(t) = v(t). As a consequence, U = V almost everywhere in (0,∞)×
RN .

We postpone the proof of Theorem 2.3 to Subsection 4.3.

Remark 2.4. It is obvious from (1.2) that the uniqueness of the solution u implies the uniqueness
of the saturated section U .

Remark 2.5. In [4], self-similar solutions are studied with the nonlinearity |u|−(1−m)u, where
0 < m < 1. It is shown that a self-similar solution cannot be continuous at t = 0 in a reasonable
way. This remains true in our case (which corresponds to m = 0). Below, we give some details.
Let p ∈ C be such that Re(p) = 2 and let u be a self-similar solution to (1.2) with profile φ.

(1) Let us define the transformation Tλ : v 7→ vλ, for any v ∈ L1
loc(RN ), when λ > 0 :

Tλ(v)( . ) = λ−pv(λ . ). The functions which satisfy this invariance property cannot be
Lq-functions in the sense that we have

Λq :=
{
v ∈ Lq(RN );∀λ > 0, Tλ(v) = v

}
=

{
0
}
,

for any q ∈ (0,∞]. Indeed, if for some q ∈ (0,∞], v ∈ Λq then a straighforward calculation
gives that

∀λ > 0, ∥v∥Lq(RN ) = λ2+
N
q ∥v∥Lq(RN ).

Therefore, v = 0. It follows that if u(0) ∈ Lq(RN ), for some 0 < q ≤ ∞, then u(0) ∈ Λq

and so necessarily u(0) = 0.
(2) It follows from above that if u ∈ C

(
[0,∞);D ′(RN )

)
is a self-similar solution to (1.2) with

u(0) ̸= 0 then for any 0 < q ≤ ∞, u /∈ C
(
[0,∞);Lq(RN )

)
. On the other hand, if for some

0 < q ≤ ∞, u ∈ C
(
(0,∞);Lq(RN )

)
then φ ∈ Lq(RN ) and it follows from (2.5) that

∀t > 0, ∥u(t)∥Lq(RN ) = t1+
N
2q ∥φ∥Lq(RN ), (2.11)



6 P. BÉGOUT, J. I. DÍAZ EJDE-2025/53

and so limt↘0 ∥u(t)∥Lq(RN ) = 0. Actually, if m ∈ {0, 1, 2}, 0 < q ≤ ∞ and φ ∈Wm,q(RN )

then by (2.5), u(t) ∈Wm,q(RN ), for any t > 0, and

∥∇u(t)∥Lq(RN ) = t
1
2+

N
2q ∥∇φ∥Lq(RN ), (2.12)

∥∂2jku(t)∥Lq(RN ) = t
N
2q ∥∂2jkφ∥Lq(RN ), (2.13)

for any t > 0 and (j, k) ∈ J1, NK2, so that limt↘0 ∥u(t)∥Wm,q(RN ) = 0 (q <∞, if m = 2).
(3) If f = 0, a ∈ R and φ has compact support then for any t ∈ R, u(t) = 0. Indeed, if g

is defined by (2.7) then g ∈ L2(RN ) and by (1.4), ∆g ∈ L2
loc(RN ). By interior elliptic

regularity, g ∈ H2
loc(RN ) (Cazenave [11, Proposition 4.1.2]). Then φ ∈ H2

loc(RN ) and since
suppφ is compact, we finally have φ ∈ H2(RN ). It follows from Lemma 5.1 below that u
satisfies the regularity (2.10). We are then allowed to take the X⋆ −X duality product of
(1.2) with iu, where X = H1(RN ) ∩ L1(RN ), to obtain that d

dt∥u(t)∥
2
L2(RN ) = 0, for any

t > 0. With help of (2.11), we then deduce that

∀t > 0, ∥φ∥L2(RN ) = ∥u(t)∥L2(RN ) = t1+
N
4 ∥φ∥L2(RN ).

Then φ = 0, from which the result follows.
(4) Assume that u(0) ̸= 0. From the structure of the self-similar solution u we easily deduce

that for any t > 0,

suppu(t) =
√
t suppφ.

Letting t↘ 0, we could conclude that suppu(0) = ∅ and then u(0) = 0. But as seen above,
u is not continuous at t = 0 in any reasonable way and we cannot infer that u(0) = 0.
Estimates on the expansion of the support of the type C

√
t were proved, for the first time,

for parabolic variational inequalities, in the paper H. Brezis and A. Friedman [10].

Remark 2.6. Let 0 < m < 1, let a ∈ A be such that Im(a) ≤ 0, let p ∈ C be such that
Re(p) = 2

1−m , and let f1, . . . , fd ∈ C
(
(0,∞);L2(RN )

)
satisfying (2.3). Assume further that for

any j ∈ J1, dK, Kj := supp fj(1) is compact, ∥fj(1)∥L2(RN ) is small enough and Kj ∩ Kℓ = ∅,
for any j ̸= ℓ. It follows from [4, Theorem 1.2] that for any j ∈ J1, dK, there exists a self-similar
solution uj to

i
∂uj
∂t

+∆uj = a|uj |−(1−m)uj + fj(t, x), (t, x) ∈ (0,∞)× RN ,

such that suppuj(1) is compact. Due to the smallness of the d norms ∥fj(1)∥L2(RN ), we also have
that for any j ̸= ℓ, suppuj(1) ∩ suppuℓ(1) = ∅. We set,

u =

d∑
j=1

uj and f =

d∑
j=1

fj .

From the structure of the self-similar solutions and since the support of the d functions uj(1) are
disjoints, we conclude that the supports of the d functions uj remain disjoints at least during some
suitable period of time (0, T ), for some T > 1. It follows that u is a self-similar solution to

i
∂u

∂t
+∆u = a|u|−(1−m)u+ f(t, x), (t, x) ∈ (0, T )× RN ,

although this equation is not linear. If m = 0, then the above arguments do not work since
we do not necessarily have that the saturated section U associated to u satisfies U = 0 when
u = 0. Nevertheless, we may still generate self-similar solutions of the evolution Schrödinger
equation (1.2), on a finite time interval (0, T ), with T > 1, having a support with more than one
connected component. Indeed, it is sufficient to work with one function f where the compactness
of f(1) and the smallness of ∥f(1)∥L2(RN ) are replaced by the following assumptions: there exist d

compact connected subsets Kj such that ∥f(1)∥L∞(Kc) is small enough, where K = ∪d
j=1Kj , and

such that for any j ̸= ℓ, Kj ∩Kℓ = ∅. We conclude with help of Theorem 2.3.



EJDE-2025/53 SATURATED SCHRÖDINGER EQUATIONS 7

Remark 2.7. It is useful to rewrite the evolution Schrödinger equation in terms of real components
of solutions and data

u = uR + iuI , f = fR + ifI ,

a = aR + iaI .

Then, the sign of the components of the coefficient a is especially crucial for understanding the
different nature of the coupled system. Theorem 2.3 holds, for instance, if a = λ− iµ with λ, µ > 0,
(in the pure elliptic system, the case of µ < 0 is also allowed: see our paper in [7]) and then we
arrive to the coupled system

∂uI
∂t

−∆uR +
λuR + µuI√
u2R + u2I

= −fR,

−∂uR
∂t

−∆uI +
λuI − µuR√
u2R + u2I

= −fI .

Here we can appreciate how this system becomes easier if we add a real coefficient to the kinetics
term (as it is the case of Ginzburg-Landau equations) since then it appears a new term ∂uR

∂t in

the first equation and a new term ∂uI

∂t in the second equation. See the paper [1].

3. Existence and uniqueness of the solutions

Definition 3.1. Let Ω ⊆ RN be an open subset, (a, b) ∈ C2 and V ∈ L∞(Ω).

(1) Let F ∈ H−1(Ω) + L∞(Ω). We shall say that a function u is a global weak solution
to (1.5) with boundary condition (1.6), if u ∈ H1

0 (Ω) ∩ L1(Ω), there is saturated section
U associated to u, and if

⟨∇u,∇v⟩L2(Ω),L2(Ω) + ⟨aU, v⟩L∞(Ω),L1(Ω) + ⟨b u, v⟩L2(Ω),L2(Ω) + ⟨V u, v⟩L2(Ω),L2(Ω)

= ⟨F, v⟩X⋆,X ,
(3.1)

for any v ∈ H1
0 (Ω) ∩ L1(Ω), where X = H1

0 (Ω) ∩ L1(Ω).
(2) Assume that Ω has a finite measure and a Lipschitz continuous boundary. Let F ∈ H1(Ω)⋆.

We shall say that a function u is a global weak solution to (1.5) with boundary condition
(1.7) if u ∈ H1(Ω), there is a saturated section U associated to u, and if (u, U) satisfies
(3.1) for any v ∈ H1(Ω), where X = H1(Ω).

Sometimes, we shall write (u, U) to designate a solution with the obvious meanings.

By convention, throughout this paper Ω denotes any open subset of RN , and (a, b) is a pair of
complex numbers. When a function will be said to satisfy the boundary condition (1.7), it will
always be assumed that Ω has a finite measure and a Lipschitz continuous boundary. Let

B = C \
{
z ∈ C; Re(z) ≤ − 1

C2
P

and Im(z) = 0
}
, (3.2)

where CP is the constant in Poincaré’s inequality (4.15) below.

Theorem 3.2 (Existence and a priori bound). Assume that |Ω| <∞ and b ∈ B. Let V ∈ L∞(Ω;R)
with V ≥ 0, a.e. in Ω. Then for any F ∈ H−1(Ω), equations (1.5)-(1.6) admit at least one global
weak solution. In addition, the symmetry property 3.3 below holds. Finally, any solution u to
(1.5)–(1.6) satisfies

∥u∥H1
0 (Ω) ≤ C, (3.3)

where C = C(∥F∥H−1(Ω), ∥V ∥L∞(Ω;R), |Ω|, |a|, |b|, N).

Property 3.3 (Symmetry Property). Furthermore, if there exists R ∈ SON (R) such that for
almost every x ∈ Ω, Rx ∈ Ω, F (Rx) = F (x) and V (Rx) = V (x) then we may construct a
solution u which also satisfies u(Rx) = u(x), for almost every x ∈ Ω. When N = 1, if Ω is
symmetric with respect to the origin and if F and V are odd functions then u is also an odd
function.



8 P. BÉGOUT, J. I. DÍAZ EJDE-2025/53

Here and in what follows, SON (R) denotes the special orthogonal group of RN . We recall that
A is defined by (2.9).

Theorem 3.4 (Existence and a priori bound). Let V ∈ L∞(Ω). Assume that a ∈ A, Im(b) ̸= 0,
Im(a) Im(b) ≥ 0 and Im(b) Im(V ) ≥ 0, a.e. in Ω. Then for any F ∈ H−1(Ω), equations (1.5)–(1.6)
admit at least one global weak solution. In addition, the symmetry property 3.3 holds. Finally,
any solution u to (1.5)–(1.6) satisfies

∥u∥2H1
0 (Ω) + ∥u∥L1(Ω) +

∫
Ω

| Im(V )||u|2dx ≤ C∥F∥2H−1(Ω), (3.4)

where C = C(∥Re(V )∥L∞(Ω), |a|, |b|). When F ∈ H1(Ω)⋆, a similar statement holds for the
boundary condition (1.7).

Remark 3.5. Note that if, in addition, Re(a) ≥ 0 and Re(ab) + Re(aV ) ≥ 0, a.e. in Ω, then the
solution given by Theorem 3.4 is unique [7, Theorem 2.8].

Theorem 3.6 (Null solution). Let V ∈ L∞(Ω). Assume that a ∈ A, Im(b) ̸= 0, Im(a) Im(b) ≥ 0
and Im(b) Im(V ) ≥ 0, a.e. in Ω. Then there exists M = M(|a|, |b|, ∥Re(V )∥L∞(Ω)) satisfying the

following property. Let F ∈ L∞(Ω) with ∥F∥L∞(Ω) ≤ |a|. If ∥F∥L∞(Ω) ≤ 1
M then the unique global

weak solution (u, U) to (1.5) with boundary condition (1.6) or (1.7) is given by

u = 0 and U =
1

a
F, (3.5)

almost everywhere in Ω.

4. Setting of the framework and proofs of the existence theorems

Let δ ∈ {0, 1} and V ∈ L∞(Ω). For n ∈ N and u ∈ L2(Ω), let

gn(u) =

{
u

|u|+(n−|u|) 1
n2
, if |u| ≤ n,

u
|u| , if |u| > n,

(4.1)

hn(u) =

{
u, if |u| ≤ n,

n u
|u| , if |u| > n,

(4.2)

fn,δ = agn(u) + (b− δ + V )hn(u). (4.3)

Let X = H1
0 (Ω) if we deal with the boundary condition (1.6), and let X = H1(Ω) if we deal with

the boundary condition (1.7). Let F ∈ X⋆. Throughout this section, u denotes any global weak
solution to

−∆u+ aU + b u+ V u = F, (4.4)

with boundary condition (1.6) or (1.7). Moreover, for each n ∈ N, un ∈ H1
0 (Ω) denotes a global

weak solution to

−∆un + fn,0(un) = F, (4.5)

with boundary condition (1.6), and vn denotes a global weak solution to

−∆vn + vn + fn,1(vn) = F, (4.6)

with boundary condition (1.6) or (1.7). Choosing as test functions u and iu in (4.4), un and iun
in (4.5), and vn and ivn in (4.6), we obtain

∥∇u∥2L2(Ω) +Re(a)∥u∥L1(Ω) +Re(b)∥u∥2L2(Ω) +

∫
Ω

Re(V )|u|2dx = ⟨F, u⟩X⋆,X , (4.7)

Im(a)∥u∥L1(Ω) + Im(b)∥u∥2L2(Ω) +

∫
Ω

Im(V )|u|2dx = ⟨F, iu⟩X⋆,X , (4.8)



EJDE-2025/53 SATURATED SCHRÖDINGER EQUATIONS 9

and for any n ∈ N,

∥∇un∥2L2(Ω) +Re(a)
(∫

{|un|≤n}

|un|2

|un|+ (n− |un|) 1
n2

dx+ ∥un∥L1({|un|>n})

)
+Re(b)

(
∥un∥2L2({|un|≤n}) + n∥un∥L1({|un|>n})

)
+

∫
{|un|≤n}

Re(V )|un|2dx+ n

∫
{|un|>n}

Re(V )|un|dx

= ⟨F, un⟩X⋆,X ,

(4.9)

Im(a)
(∫

{|un|≤n}

|un|2

|un|+ (n− |un|) 1
n2

dx+ ∥un∥L1({|un|>n})

)
+ Im(b)

(
∥un∥2L2({|un|≤n}) + n∥un∥L1({|un|>n})

)
+

∫
{|un|≤n}

Im(V )|un|2dx+ n

∫
{|un|>n}

Im(V )|un|dx

= ⟨F, iun⟩X⋆,X ,

(4.10)

∥vn∥2X +Re(a)
(∫

{|vn|≤n}

|vn|2

|vn|+ (n− |vn|) 1
n2

dx+ ∥vn∥L1({|vn|>n})

)
≤

(
|Re(b)|+ 1 + ∥Re(V )∥L∞(Ω)

)(
∥vn∥2L2({|vn|≤n}) + n∥vn∥L1({|vn|>n})

)
+ ⟨F, vn⟩X⋆,X ,

(4.11)

and

vn satisfies (4.10). (4.12)

We note that for each w ∈ L2(Ω) and n ∈ N, we have that∫
{|w|≤n}

|w|2

|w|+ (n− |w|) 1
n2

dx+ ∥w∥L1({|w|>n}) ≤ ∥w∥L1(Ω), (4.13)

∥w∥2L2({|w|≤n}) + n∥w∥L1({|w|>n}) ≤ ∥w∥2L2(Ω). (4.14)

Finally, we recall that if |Ω| <∞ then we have Poincaré’s inequality:

∀w ∈ H1
0 (Ω), ∥w∥L2(Ω) ≤ CP∥∇w∥L2(Ω), (4.15)

where CP = CP(|Ω|, N), and then

∀w ∈ H1
0 (Ω), ∥w∥L1(Ω) ≤ |Ω| 12 ∥w∥L2(Ω) ≤ CP|Ω|

1
2 ∥∇w∥L2(Ω), (4.16)

∀w ∈ H1
0 (Ω), ∥w∥H1

0 (Ω) ≤ (1 + CP)∥∇w∥L2(Ω). (4.17)

4.1. Homogeneous Dirichlet boundary condition with a domain of finite measure.
Throughout this subsection, we deal with the boundary condition (1.6) and assume that |Ω| <∞.

Lemma 4.1. If Re(b) ≥ 0 and Re(V ) ≥ 0 then

∥∇u∥L2(Ω) + ∥∇un∥L2(Ω) +

∫
Ω

Re(V )|u|2dx ≤ C(∥F∥H−1(Ω), |Ω|, |Re(a)|, N), (4.18)

for any n ∈ N.

Proof. Starting with (4.9) and using (4.13)–(4.17), we obtain for any n ∈ N,

∥∇un∥2L2(Ω) ≤
(
|Re(a)|CP|Ω|

1
2 + (1 + CP)∥F∥H−1(Ω)

)
∥∇un∥L2(Ω),

from which the result follows for ∥∇un∥L2(Ω). Starting with (4.7), we obtain the estimate for

∥∇u∥2L2(Ω) +
∫
Ω
Re(V )|u|2dx in the same way. □

Lemma 4.2. If − 1
C2

P
< Re(b) < 0 and Re(V ) ≥ 0 then u and (un)n∈N satisfy (4.18).



10 P. BÉGOUT, J. I. DÍAZ EJDE-2025/53

Proof. Starting with (4.9) and using (4.13), (4.14), (4.16) and (4.17), we obtain that for any n ∈ N,

∥∇un∥2L2(Ω) ≤ (|Re(a)|CP|Ω|
1
2 + (1 + CP)∥F∥H−1(Ω))∥∇un∥L2(Ω) − Re(b)C2

P∥∇un∥2L2(Ω),

from which we obtain

(1 + Re(b)C2
P)∥∇un∥L2(Ω) ≤ (|Re(a)|CP|Ω|

1
2 + (1 + CP)∥F∥H−1(Ω)),

for any n ∈ N. But 1 + Re(b)C2
P > 0 and then the result follows for ∥∇un∥L2(Ω). Starting with

(4.7), we obtain the estimate for ∥∇u∥2L2(Ω) +
∫
Ω
Re(V )|u|2dx in the same way. □

Lemma 4.3. If Im(b) ̸= 0 and Im(b) Im(V ) ≥ 0 then for any n ∈ N,

∥∇u∥L2(Ω) + ∥∇un∥L2(Ω) +

∫
Ω

| Im(V )||u|2dx ≤ C,

where C = C(∥F∥H−1(Ω), ∥Re(V )∥L∞(Ω), |Ω|, |a|, |b|, N).

Proof. Let n ∈ N. Since Im(b) ̸= 0 and Im(b) Im(V ) ≥ 0, we infer from (4.10) with help of (4.13),
(4.16) and (4.17) that

| Im(b)|
(
∥un∥2L2({|un|≤n}) + n∥un∥L1({|un|>n})

)
≤ C1∥∇un∥L2(Ω), (4.19)

where

C1 = | Im(a)||Ω| 12CP + (1 + CP)∥F∥H−1(Ω).

It follows that ∫
{|un|≤n}

|Re(V )||un|2dx+ n

∫
{|un|>n}

|Re(V )||un|dx

≤ ∥Re(V )∥L∞(Ω)

(
∥un∥2L2({|un|≤n}) + n∥un∥L1({|un|>n})

)
≤ C1| Im(b)|−1∥Re(V )∥L∞(Ω)∥∇un∥L2(Ω).

This yields with (4.9), (4.13), (4.16), (4.17) and (4.19) that

∥∇un∥2L2(Ω) ≤
(
CP|Re(a)||Ω|

1
2 + C1| Im(b)|−1(|Re(b)|+ ∥Re(V )∥L∞(Ω))

+ (1 + CP)∥F∥H−1(Ω)

)
∥∇un∥L2(Ω),

which gives the desired result for ∥∇un∥L2(Ω). For ∥∇u∥L2(Ω), we proceed as follows. Using (4.8)
in place of (4.13), we obtain in the same way as for (4.19) that

| Im(b)|∥u∥2L2(Ω) +

∫
Ω

| Im(V )||u|2dx ≤ C2∥∇u∥L2(Ω), (4.20)

where

C2 = | Im(a)||Ω| 12CP + (1 + CP)∥F∥H−1(Ω).

As a consequence, ∫
Ω

|Re(V )||u|2dx ≤ C2| Im(b)|−1∥Re(V )∥L∞(Ω)∥∇u∥L2(Ω). (4.21)

Using (4.16), (4.17), (4.20) and (4.21) in (4.7), we obtain that

∥∇u∥2L2(Ω) ≤
(
CP|Re(a)||Ω|

1
2 + C2| Im(b)|−1(|Re(b)|+ ∥Re(V )∥L∞(Ω))

+ (1 + CP)∥F∥H−1(Ω)

)
∥∇u∥L2(Ω).

Hence the result with the help of (4.20). □



EJDE-2025/53 SATURATED SCHRÖDINGER EQUATIONS 11

4.2. General case. In this subsection, we deal with both boundary conditions (1.6) and (1.7).
In addition, no assumption about the open set Ω is made. We recall that X = H1

0 (Ω) if we deal
with the boundary condition (1.6), and X = H1(Ω) if we deal with the boundary condition (1.7).
Let F ∈ X⋆.

Lemma 4.4. If a ∈ A, Im(b) ̸= 0, Im(a) Im(b) ≥ 0 and Im(b) Im(V ) ≥ 0 then

∥u∥2X + ∥u∥L1(Ω) +

∫
Ω

| Im(V )||u|2dx ≤ C(|⟨F, iu⟩X⋆,X |+ |⟨F, u⟩X⋆,X |), (4.22)

∥vn∥2X +

∫
{|vn|≤n}

|vn|2

|vn|+ (n− |vn|) 1
n2

dx ≤ C∥F∥2X⋆ , (4.23)

for any n ∈ N, where C = C(∥Re(V )∥L∞(Ω), |a|, |b|).

Proof. Let n ∈ N. By our assumptions, (4.12) may be written as

| Im(a)|
(∫

{|vn|≤n}

|vn|2

|un|+ (n− |vn|) 1
n2

dx+ ∥vn∥L1({|vn|>n})

)
+ | Im(b)|

(
∥vn∥2L2({|vn|≤n}) + n∥vn∥L1({|vn|>n})

)
+

∫
{|vn|≤n}

| Im(V )||un|2dx+ n

∫
{|vn|>n}

| Im(V )||vn|dx

= |⟨F, ivn⟩X⋆,X |,

(4.24)

If Re(a) > 0 then by (4.11) and (4.24), we have

∥vn∥2X +Re(a)

∫
{|vn|≤n}

|vn|2

|vn|+ (n− |vn|) 1
n2

dx

≤
( |Re(b)|+ 1 + ∥Re(V )∥L∞(Ω)

| Im(b)|

)
|⟨F, ivn⟩X⋆,X |+ |⟨F, vn⟩X⋆,X |.

If Re(a) ≤ 0 then Im(a) ̸= 0. Multiplying (4.24) by L := |Re(a)|+1
| Im(a)| and adding the result to (4.11),

we obtain that

∥vn∥2X +

∫
{|vn|≤n}

|vn|2

|vn|+ (n− |vn|) 1
n2

dx

≤
( |Re(b)|+ 1 + ∥Re(V )∥L∞(Ω)

| Im(b)|
+ L

)
|⟨F, ivn⟩X⋆,X |+ |⟨F, vn⟩X⋆,X |.

In both cases, we obtain that

∥vn∥2X +

∫
{|vn|≤n}

|vn|2

|un|+ (n− |vn|) 1
n2

dx ≤ C(|⟨F, ivn⟩X⋆,X |+ |⟨F, vn⟩X⋆,X |),

for some C = C(∥Re(V )∥L∞(Ω), |a|, |b|). Applying Young’s inequality to the above, we obtain
(4.23). Using (4.7) and (4.8) instead of (4.11) and (4.12), we obtain (4.22) in the same way. □

4.3. Proofs of the existence and compactness theorems.

Proof of Theorems 3.2 and 3.4. We first note that (3.3) comes from Lemmas 4.1–4.3 and (4.15),
and that (3.4) comes from Lemma 4.4 and Young’s inequality. It remains to establish the existence
part of the theorems. We first assume that |Ω| <∞. Let F be as in the theorems. For each n ∈ N,
let un be a global weak solution to (4.5) and (1.6), and let vn be a global weak solution to (4.6) and
(1.6) (respectively, to (4.6) and (1.7)). Indeed, such solutions exist with the help of [7, Lemma 6.5].
By Lemmas 4.1–4.3, (4.13), (4.15) and (4.16), it follows that (un)n∈N is bounded in H1

0 (Ω) and( |un|2

|un|+ (n− |un|) 1
n2

1{|un|≤n}

)
n∈N

is bounded in L1(Ω).



12 P. BÉGOUT, J. I. DÍAZ EJDE-2025/53

By [7, Lemma 6.2], we may extract a subsequence of (un)n∈N which converges to a solution of (1.5)–
(1.6). Theorem 3.2 is then proved. By Lemma 4.4, (vn)n∈N is bounded in H1

0 (Ω) (respectively, in
H1(Ω)) and ( |vn|2

|vn|+ (n− |vn|) 1
n2

1{|vn|≤n}

)
n∈N

is bounded in L1(Ω).

By [7, Lemma 6.2], (respectively, [7, Lemma 6.3],) we may extract a subsequence of (vn)n∈N
which converges to a solution of (1.5)–(1.6) (respectively, (1.5) and (1.7)). This completes the
proof of Theorem 3.2, then Theorem 3.4 is proved in the case |Ω| < ∞. To complete the proof,
it remains to show that (1.5)–(1.6) admits a solution when |Ω| = ∞. An appeal to (3.4) and
the Extension Lemma ([7, Lemma 6.9] applied with Ωn = Ω ∩ B(0, n)) gives the existence of
a u ∈ H1

0 (Ω) and of a saturated section U associated to u such that (u, U) satisfies (1.5) in
D ′(Ω). But ∆u, V u, F ∈ H−1(Ω) and U ∈ L∞(Ω) so that the equation (1.5) makes sense in
H−1(Ω) + L∞(Ω) ↪→ D ′(Ω). Theorem 3.4 is then proved. □

Proof of Theorem 3.6. We indeed check that (u, U) defined by (3.5) is a solution to (1.5). Now,
assume that (u, U) is a solution to (1.5). Taking the duality product of (1.5) with u and iu, we
have that

∥∇u∥2L2(Ω) +Re(a)∥u∥L1(Ω) + (Re(b)− ∥V ∥L∞(Ω))∥u∥2L2(Ω) ≤
∫
Ω

|Fu|dx, (4.25)

| Im(a)|∥u∥L1(Ω) + | Im(b)|∥u∥2L2(Ω) +

∫
Ω

| Im(V )||u|2dx ≤
∫
Ω

|Fu|dx. (4.26)

Since we have either Re(a) > 0 or | Im(a)| > 0, and since | Im(b)| > 0, we may find a C =
C(|a|, |b|, ∥Re(V )∥L∞(Ω)) such that Re(a) +C| Im(a)| > 0 and Re(b)−∥V ∥L∞(Ω) +C| Im(b)| ≥ 1.
We then multiply (4.26) by C and sum the result to (4.25). This yields

∥u∥2H1(Ω) + ∥u∥L1(Ω) ≤M

∫
Ω

|Fu|dx,

for some M = M(|a|, |b|, ∥Re(V )∥L∞(Ω)). Applying Hölder’s inequality to the above, we obtain
that

∥u∥2H1(Ω) + (1−M∥F∥L∞(Ω))∥u∥L1(Ω) ≤ 0.

Hence (3.5) if ∥F∥L∞(Ω) ≤ 1
M . □

Proof of Theorem 2.3. Let K be a compact subset of RN for which F|Kc ∈ L∞(Kc). Let R > 0

be such that K ⊂ B(0, R) and let ε ∈ (0, 1).

Proof of property (1). Let us write (1.4) as

−∆g + aG+ bg + V g = F1, (4.27)

where b = −iN+2p
4 , V (x) = − 1

16 |x|
2 and F1 = −Fe−i

|x|2
8 . We have that Im(b) = −N+4

4 < 0,

Im(a) Im(b) ≥ 0 and Im(b) Im(V ) = 0, in RN . It follows that (4.27) falls into the scope of Theo-
rem 3.4 and then (4.27) admits a solution gε ∈ H1

0 (B(0, R+2ε)), where the right member of (4.27)
is F1|B(0,R+2ε). By global elliptic regularity gε ∈ H2(B(0, R + 2ε)), (Gilbarg and Trudinger [16,
Theorem 8.12, p.186]). Let us denote by Gε the saturated section associated to gε. Applying [5,
Theorem 3.1], we have that

∥∇gε∥2L2(BR,ε,x0
(ρ)) +Re(a)∥gε∥L1(BR,ε,x0

(ρ)) +Re(b)∥gε∥2L2(BR,ε,x0
(ρ))

−
∫
BR,ε,x0

(ρ)

|x|2

16
|gε|2dx

= Re
(∫

BR,ε,x0
(ρ)

F1 gε dx
)
+Re

(∫
SR,ε,x0

(ρ)

gε∇gε.
x− x0
|x− x0|

dσ
)
,

(4.28)

| Im(a)|∥gε∥L1(BR,ε,x0
(ρ)) + | Im(b)|∥gε∥2L2(BR,ε,x0

(ρ))

= − Im
(∫

BR,ε,x0
(ρ)

F1 gε dx
)
− Im

(∫
SR,ε,x0

(ρ)

gε∇gε.
x− x0
|x− x0|

dσ
)
,

(4.29)



EJDE-2025/53 SATURATED SCHRÖDINGER EQUATIONS 13

for any x0 ∈ B(0, R + 2ε) and ρ ∈ [0, 2ε), where BR,ε,x0
(ρ) = B(0, R + 2ε) ∩ B(x0, ρ) and

SR,ε,x0
(ρ) = B(0, R+2ε)∩S(x0, ρ). Let us denote by g ∈ H1(RN ) the extension by 0 of gε outside

of B(0, R + 2ε). Since we have either Re(a) > 0 or | Im(a)| > 0, and | Im(b)| > 0, we may find a

C = C(|a|, | Im(p)|, R,N) such that Re(a)+C| Im(a)| > 0 and Re(b)− (R+2)2

16 +C| Im(b)| ≥ 1. We
then multiply (4.29) by C and sum the result to (4.28). This yields

∥g∥2H1(B(x0,ρ))
+ ∥g∥L1(B(x0,ρ)) ≤ C1

(∫
B(x0,ρ)

|F1g|+
∣∣∣ ∫

S(x0,ρ)

g∇g. x− x0
|x− x0|

dσ
∣∣∣),

for any x0 ∈ B(0, R + 2ε) and ρ ∈ [0, 2ε), and for some C1 = C1(|a|, | Im(p)|, R,N). It follows
from Hölder’s inequality that for M ≥ 2C1, if ∥F∥L∞(Kc) ≤ 1

M then

∥g∥2H1(B(x0,ρ))
+ ∥g∥L1(B(x0,ρ)) ≤M

∣∣∣ ∫
S(x0,ρ)

g∇g. x− x0
|x− x0|

dσ
∣∣∣, (4.30)

for any x0 ∈ B(0, R + 2ε) and ρ ∈ [0, 2ε) such that K ∩ B(x0, 2ε) = ∅. It follows from [7,
Theorem 4.1] that there exists ρmax ≥ 0 such that g = 0, a.e. in B(x0, ρmax), for any x0 ∈
B(0, R + 2ε) such that K ∩ B(x0, 2ε) = ∅. By (3.4) and [7, Theorem 4.1], there exists δ =
δ(|a|, | Im(p)|, R, ε,N) such that if ∥F∥L2(RN ) ≤ δ then ρmax > ε. We then deduce that g = gε = 0,

a.e. in B(0, R + 2ε) \ K(ε). Now, let us define G on RN by G = Gε, in B(0, R + 2ε) and by

G = − 1
aFe

−i
|x|2
8 , in B(0, R + 2ε)c. Choosing also M ≥ |a|−1, it follows that G is a saturated

section associated to g. So, we have shown that (g,G) is a solution to (1.4), g ∈ H2(RN ) and
supp g ⊂ K(ε). Now, we define φ and Φ by (2.7) and (2.8), respectively, and finally, u and U by
(2.5) and (2.6), respectively. The proof of (2.10) comes from standard arguments of integration
theory, but for the convenience of the reader, we postpone its proof to the Appendix 5. This
completes the proof.

Proof of property (2). Using the change of variables (2.7) and (2.8), we are brought back to
show the uniqueness for the equation (1.4). In both cases (2)(a) and (2)(b), φ and ϕ belong to
L2(RN ) and are compactly supported. It follows that the corresponding solutions to (1.4) belong
to L2(RN ) and their Laplacian belongs to L2

loc(RN ). By interior elliptic regularity, they belong to
H2

loc(RN ) (Cazenave [11, Proposition 4.1.2]). Since they are compactly supported, they actually
belong to H2(RN ) and it is sufficient to show the uniqueness for (1.4) set in B(0, r), where r > 0
is large enough to have suppφ ∪ suppϕ ⊂ B(0, r). It follows that (1.4) falls into the scope of the
uniqueness [7, Theorem 2.8]. Since also a ∈ A and Re(a) ≥ 0, we only have to show that

Re(ab) + Re(aV ) > 0, a.e. in B(0, r),

where b and V are as in (4.27). If Re(a) = 0 then Im(a) < 0 and Re(ab)+Re(aV ) = − Im(a)N+4
4 >

0, over RN . If Re(a) > 0 then

Re(ab) + Re(aV ) =
1

2
Re(a) Im(p)− Im(a)

N + 4

4
− 1

16
Re(a)|x|2, in RN .

Using (2)(b), we have that

Re(ab) + Re(aV ) >
Re(a)

16

(
8 Im(p)− 4

Im(a)

Re(a)
(N + 4)− r2

)
≥ 0, in B(0, r).

This concludes the proof of the theorem. □

5. Appendix

Lemma 5.1. Let m ∈ N0, 1 < q < ∞ and φ ∈ Wm,q(RN ). Let p ∈ C, and let u be defined by
(2.5). Then

u ∈ C
(
(0,∞);Wm,q(RN )

)
. (5.1)

If, in addition, suppφ is compact and m ≥ 1 then

u ∈ ∩m
j=1C

j
(
(0,∞);Wm−j,q(RN )

)
. (5.2)



14 P. BÉGOUT, J. I. DÍAZ EJDE-2025/53

Proof. Let 1 < q < ∞, p ∈ C, φ ∈ Lq(RN ), and u be defined by (2.5). Let t > 0. Let

(tn)n∈N ⊂ (0,∞) be such that tn
n→∞−−−−→ t. We claim that

u(tn)⇀ u(t) in Lq(Ω)w as n→ ∞ (5.3)

By (2.11), (u(tn))n∈N is bounded in Lq(RN ). So, it is enough to show that u(tn) → u(t) in D ′(RN )
as t→ ∞. Let θ ∈ D(RN ). By change of variables, we have for any n ∈ N,

⟨u(tn), θ⟩D′(RN ),D(RN ) = Re

∫
RN

t
p+N

2
n φ(x)θ(

√
tnx)dx,

⟨u(t), θ⟩D′(RN ),D(RN ) = Re

∫
RN

t
p+N

2 φ(x)θ(
√
tx)dx.

It follows from the dominated convergence Theorem that

⟨u(tn), θ⟩D′(RN ),D(RN )
n→∞−−−−→ ⟨u(t), θ⟩D′(RN ),D(RN ).

from which we obtain (5.3). By (2.11), we also have that

∥u(tn)∥Lq(RN )
n→∞−−−−→ ∥u(t)∥Lq(RN ). (5.4)

By (5.3), (5.4) and the uniform convexity of the Lq-spaces, we infer that

u(tn)
Lq(RN )−−−−−→
n→∞

u(t),

proving that u ∈ C
(
(0,∞);Lq(RN )

)
. Now assume that φ ∈ Wm,q(RN ), for some m ∈ N. Then

(5.1) follows with the same arguments. We have for any n ∈ N and almost every x ∈ RN ,

∂u

∂t
(tn, x) =

p

2
t
p−2
2

n φ
( x√

tn

)
− 1

2
t
p−3
2

n x.∇φ
( x√

tn

)
,

∂u

∂t
(t, x) =

p

2
t
p−2
2 φ

( x√
t

)
− 1

2
t
p−3
2 x.∇φ

( x√
t

)
.

If suppφ is compact, then we may proceed as above to show that

p

2
t
p−2
2

n φ
( ·√

tn

)
Lq(RN )−−−−−→
n→∞

p

2
t
p−2
2 φ

( ·√
t

)
,

1

2
t
p−3
2

n (·).∇φ
( ·√

tn

)
Lq(RN )−−−−−→
n→∞

1

2
t
p−3
2 (·).∇φ

( ·√
t

)
.

As a consequence, ∂u
∂t (tn)

Lq(RN )−−−−−→
n→∞

∂u
∂t (t) and then u ∈ C1

(
(0,∞);Lq(RN )

)
. The other regularities

in (5.2) are obtained in the same way, and the details are left to the reader. □

Acknowledgements. P. Bégout acknowledges funding from ANR under grant ANR-17-EUR-
0010 (Investissements d’Avenir program) J. I. Dı́az was partially supported by project PID2023-
146754 NB-I00 funded by MCIU/AEI/10.13039/501100011033 and FEDER, EU. MCIU/AEI/
10.13039/-501100011033/FEDER, EU.

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Unabridged republication of the 1967 original.

Pascal Bégout
Toulouse School of Economics, Université Toulouse Capitole, Institut de Mathématiques de Toulouse,

1, Esplanade de l’Université, 31080 Toulouse Cedex 6, France

Email address: Pascal.Begout@math.cnrs.fr, ORCID: 0000-0002-9172-3057

Jesús Ildefonso D́ıaz

Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Plaza de las Ciencias,
3, 28040 Madrid, Spain

Email address: jidiaz@ucm.es, ORCID: 0000-0003-1730-9509


	1. Introduction
	2. Self-similar solutions
	3. Existence and uniqueness of the solutions
	4. Setting of the framework and proofs of the existence theorems
	4.1. Homogeneous Dirichlet boundary condition with a domain of finite measure
	4.2. General case
	4.3. Proofs of the existence and compactness theorems

	5. Appendix
	Acknowledgements

	References