Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 28, pp. 1–11.

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

WEAKLY MONOTONE DECREASING SOLUTIONS TO

ELLIPTIC SCHRÖDINGER INTEGRAL SYSTEMS

EDWARD CHERNYSH

Abstract. In this article, we study positive solutions to an elliptic Schrödinger
system in Rn for n ≥ 2. We give general conditions guaranteeing the non-

existence of positive solutions and introduce weakly monotone decreasing func-

tions. We also establish lower-bounds on the decay rates of positive solutions
and obtain upper-bounds when these are weakly monotone decreasing.

1. Introduction and main results

In this article, we investigate positive solutions to the elliptic Schrödinger integral
system

u(x) =

∫
Rn

φ(y)u(y)rv(y)q + Γ1(y, u, v)

|x− y|n−α|y|σ1
dy, x ∈ Rn,

v(x) =

∫
Rn

ψ(y)u(y)pv(y)s + Γ2(y, u, v)

|x− y|n−α|y|σ2
dy, x ∈ Rn,

(1.1)

where

n ≥ 2, α ∈ (0, n), p, q, r, s ≥ 0, r, s ∈ [0, 1], σ1, σ2 ∈ (−∞, α). (1.2)

We assume that φ, ψ,Γ1 and Γ2 are non-negative in their arguments and that

lim inf
|x|→∞

φ(x) > 0 and lim inf
|x|→∞

ψ(x) > 0. (1.3)

These integral systems are closely related, and equivalent under the appropriate
regularity and decay assumptions (see Vétois [3] and Villavert [4, 5] for results
regarding this relationship), to differential equations of the form

(−∆)α/2u(x) ≡ (φ(x)v(x)qu(x)r + Γ1(x, u, v))|x|−σ1 ,

(−∆)α/2v(x) ≡ (ψ(x)u(x)pv(x)s + Γ2(x, u, v))|x|−σ2

with x ∈ Rn \ {0}. Systems of the form in (1.1) arise in nonlinear optics and
in the modelling of Bose-Einstein double condensates (consult Vétois [3] and the
references therein). It is also worth noting that Schrödinger equations in the whole
Rn with Γ1,Γ2 ≡ 0 and φ ≡ ψ ≡ 1 are central in the blow-up analysis of solutions
to more general equations on manifolds and domains in Rn. Furthermore, a priori

2010 Mathematics Subject Classification. 35J60, 45G15, 35B45.
Key words and phrases. Elliptic Schrödinger system; poly-harmonic equation;

a priori decay estimate; weakly monotone decreasing solution.
c©2021 Texas State University.

Submitted May 4, 2019. Published April 13, 2021.

1



2 E. CHERNYSH EJDE-2021/28

decay estimates for solutions of (1.1) are useful in establishing the symmetry of
solutions (see, for instance, Liu-Ma [2] and Vétois [3]).

When obtaining a priori estimates, it is common to consider decay solutions, i.e.
solution pairs (u, v) such that u(x) ' |x|−θ1 and v(x) ' |x|−θ2 , for some θ1, θ2 > 0.
Here, u(x) ' |x|−θ means that there exists a constant C > 0 such that

1

C
|x|−θ ≤ u(x) ≤ C|x|−θ, as |x| → ∞.

This decay assumption was made in Villavert [4] when considering positive bounded
solutions to the Hardy-Sobolev type system

u(x) =

∫
R

v(y)q

|x− y|n−α|y|σ1
dy,

v(x) =

∫
R

u(y)p

|x− y|n−α|y|σ2
dy

(1.4)

with σ1, σ2 ∈ [0, α). We now introduce the notion of a weakly monotone decreasing
function, which extends the concept of a decay solution.

Definition 1.1. A measurable function f : Rn → (0,∞] is said to be weakly mono-
tone decreasing provided f is finite almost everywhere and there exist constants
C,R > 0 such that f(x) ≤ Cf(y) whenever |x| ≥ |y| ≥ R.

Remark 1.2. If f is weakly monotone decreasing, then {f = ∞} must also be
bounded.

The set of all weakly monotone decreasing functions shall henceforth be denoted
by Wm(Rn). It is not difficult to see that all decay functions are weakly monotone
decreasing. Thus, it is natural to view weakly monotone decreasing functions as
a generalization of decay solutions. This notion of weak monotonicity will play a
crucial role when deducing upper-bounds on the decay rates of positive solutions
to (1.1).

Let us now define two positive constants that play a fundamental role in our
asymptotic analysis:

r0 :=
p(α− σ1) + (α− σ2)(1− r)

pq − (1− s)(1− r)
,

s0 :=
q(α− σ2) + (α− σ1)(1− s)

pq − (1− s)(1− r)
.

Recall that we use the notation f(x) . g(x) to state that there exists C,R > 0
such that f(x) ≤ Cg(x) for all x satisfying |x| ≥ R.

Theorem 1.3. Suppose that (1.2)-(1.3) hold and let (u, v) be a positive solution
pair to (1.1). Then

u(x) &

{
(1 + |x|)−min{n−α,(q+r)(n−α)−(α−σ1)}, (q + r)(n− α) 6= n− σ1,
(1 + |x|)−(n−α) ln(1 + |x|), (q + r)(n− α) = n− σ1

(1.5)

and

v(x) &

{
(1 + |x|)−min{n−α,(p+s)(n−α)−(α−σ2)}, (p+ s)(n− α) 6= n− σ2,
(1 + |x|)−(n−α) ln(1 + |x|), (p+ s)(n− α) = n− σ2.

(1.6)



EJDE-2021/28 ELLIPTIC SCHRÖDINGER SYSTEMS 3

Suppose, in addition, that u and v are weakly monotone decreasing. If

pq > (1− r)(1− s),
then

u(x) . |x|−s0 and v(x) . |x|−r0 . (1.7)

In several cases, the lower and upper estimates obtained in Theorem 1.3 are
known to be sharp. Villavert [4] showed that all integrable solutions (u, v) to (1.4)
decay precisely with the rates in (1.5)-(1.6). The lower bounds are also known to be
optimal in the case r = s = σ1,2 = 0, Γ1 ≡ Γ2 ≡ 0 and φ ≡ ψ ≡ 1 (see Vétois [3]).
The bounds in (1.5)-(1.6) were also found to be sharp for positive C2(Rn) radially
symmetric solutions of the equation ∆u + K(x)up ≡ 0, under suitable conditions
for K and p (the reader may consult Li [1] for more details). In fact, Li [1] also
showed that these radial C2(Rn) solutions to ∆u+K(x)up ≡ 0 decay with the rates
in (1.7) when u 6' |x|2−α. We also point out that the upper-bound estimates in
(1.7) were obtained in Villavert [4] for bounded decay solutions to (1.4). Moreover,
in Villavert [4] it was also established that the estimates in (1.7) are sharp for all
non-integrable decay solutions to (1.4).

The first section is devoted to the proof of Theorem 1.3. In the second section, we
shall instead give conditions under which no positive or weakly monotone decreasing
solution pairs to (1.1) can exist. We also provide bounds on the weighting terms
σ1 and σ2 required for the existence of solutions. These are contained within the
following theorem.

Theorem 1.4. Assume (1.2)-(1.3) hold. System (1.1) admits no positive solutions
if either pq = 0,

σ1 ≤ α− (q + r)(n− α), or σ2 ≤ α− (p+ s)(n− α).

Furthermore, no weakly monotone decreasing solutions exist if pq ≤ (1− r)(1− s).

2. Decay estimates

For the entirety of this section, we assume that u and v are positive functions
defined on Rn and that (1.2)-(1.3) hold. We begin by deriving a priori upper-bound
estimates for weakly monotone decreasing solution pairs. For the remainder of this
paper, we denote by meas(·) the Lebesgue measure on Rn.

Proposition 2.1. Let (u, v) be a positive weakly monotone decreasing solution pair
to (1.1). If pq > (1− r)(1− s), then

u(x) . |x|−s0 and v(x) . |x|−r0 .

Proof. We shall follow the strategy illustrated in Villavert [4]. Since u and v are
both weakly monotone decreasing, we are free to choose positive constants R and
C such that u and v satisfy

Cu(x) ≤ u(y) and Cv(x) ≤ v(y)

whenever |x| ≥ |y| ≥ R. Moreover, by invoking (1.3), we are free to assume that

min{φ(x), ψ(x)} ≥ γ0 > 0, ∀|x| ≥ R,
where γ0 is a constant. For |x| ≥ 2R we define an annulus in space

Ax := {y ∈ Rn :
|x|
2
< |y| < |x|}



4 E. CHERNYSH EJDE-2021/28

and deduce from the non-negativity of u and v that, for all such x,

u(x) ≥
∫
Rn

φ(y)v(y)qu(y)r

|x− y|n−α|y|σ1
dy ≥ γ0

∫
Ax

v(y)qu(y)r

|x− y|n−α|y|σ1
dy.

Now, using that both u and v are weakly monotone decreasing, we find (after a
correction of the constant C)

u(x) ≥ C
∫
Ax

v(x)qu(x)r

|x− y|n−α|y|σ1
dy

≥ Cu(x)rv(x)q|x|α−n
∫
Ax

1

|y|σ1
dy

≥ Cu(x)rv(x)q|x|α−n−σ1 meas(Ax),

where we have used that |x− y| ≤ 2|x| and |y| ≤ |x|. Since

meas(Ax) = c
(
|x|n − |x|

n

2n
)

for a constant c > 0, it follows that

u(x) ≥ Cu(x)rv(x)q|x|α−σ1 , as |x| → ∞. (2.1)

By symmetry of the system, a verbatim argument yields

v(x) ≥ Cu(x)pv(x)s|x|α−σ2 , as |x| → ∞. (2.2)

We now distinguish two possible cases.

Case 1: r, s ∈ [0, 1). Using (2.1) and (2.2) we have, as |x| → ∞,

u(x) ≥ Cv(x)
q

1−r |x|
α−σ1
1−r and v(x) ≥ Cu(x)

p
1−s |x|

α−σ2
1−s .

Combining these inequalities yields, for |x| large,

u(x) ≥ Cu(x)
pq

(1−s)(1−r) |x|
q(α−σ2)

(1−s)(1−r)+
α−σ1
1−r .

The above implies that, as |x| → ∞,

u(x)
pq−(1−s)(1−r)

(1−s)(1−r) ≤ C|x|−
q(α−σ2)+(α−σ1)(1−s)

(1−s)(1−r) .

Consequently, as |x| → ∞

u(x) ≤ C|x|−
q(α−σ2)+(α−σ1)(1−s)

pq−(1−s)(1−r) = C|x|−s0 .
A symmetric argument shows that v(x) . |x|−r0 as well.

Case 2: r = 1 or s = 1. We may assume without loss of generality that r = 1. We
invoke equation (2.1) to find that, after a correction of C,

v(x) ≤ C|x|−
α−σ1
q = C|x|−r0 , as |x| → ∞. (2.3)

Similarly, if s = 1 we use (2.2) and take roots to obtain

u(x) ≤ C|x|−
α−σ2
p = C|x|−s0 , as |x| → ∞.

On the other hand, if 0 ≤ s < 1, it follows from (2.3) that for all suitably large x,

v(x)1−s ≤ C|x|−
(α−σ1)(1−s)

q .

Combining the above estimate with (2.2) grants us the following, which is valid for
all |x| large,

Cu(x)p|x|α−σ2 ≤ v(x)1−s ≤ C ′|x|−
(α−σ1)(1−s)

q



EJDE-2021/28 ELLIPTIC SCHRÖDINGER SYSTEMS 5

whence we have

u(x)p ≤ C|x|−
q(α−σ2)+(α−σ1)(1−s)

q , as |x| → ∞.
Taking roots we obtain

u(x) ≤ C|x|−
q(α−σ2)+(α−σ1)(1−s)

pq = C|x|−s0 , as |x| → ∞.
A verbatim argument applies to the case of s = 1 and 0 ≤ r < 1. This completes
the proof. �

Lemma 2.2. Let (u, v) be a positive solution pair to (1.1). Then

min{u(x), v(x)} & 1

(1 + |x|)n−α
. (2.4)

Proof. By (1.3), we may choose R > 0 such that

min{φ(x), ψ(x)} ≥ γ0 > 0

whenever |x| ≥ R− 1. Once again, we define an annulus in Rn

A := {y ∈ Rn : R− 1 < |y| < R}.
Let x ∈ Rn be such that |x| ≥ R and let y ∈ A. Then |x− y| ≤ |x|+R, whence

u(x) ≥ γ0
∫
A

v(y)qu(y)r

|x− y|n−α|y|σ1
dy ≥ C

(R+ |x|)n−α

∫
A

v(y)qu(y)r

|y|σ1
dy.

By taking x such that u(x) <∞, it follows that
∫
A
v(y)qu(y)r

|y|σ1 dy is a finite positive

constant independent of x, thereby yielding the desired inequality for u. By a
symmetric argument, the same inequality holds true for v. �

We are now capable of proving our generalized version of Villavert [4, THM-1].

Proof of Theorem 1.3. We shall prove this result in two steps. The first establishes
lower bounds for all positive solutions and the second step gives a sharper estimate
on positive solutions in the cases

(q + r)(n− α) = n− σ1 and (p+ s)(n− α) = n− σ2.
Step 1. Suppose u and v are positive solutions to (1.1). Then

u(x) & (1 + |x|)−min{n−α,(q+r)(n−α)−(α−σ1)},

v(x) & (1 + |x|)−min{n−α,(p+s)(n−α)−(α−σ2)}.

Proof of Step 1. For |x| > 0 we define an open ball

Bx := {y ∈ Rn : |x− y| < |x|
2
},

and observe that by letting |x| → ∞, we can make y ∈ Bx arbitrarily large. Thus,
by Lemma 2.2, as |x| → ∞ we have (letting γ0 be the same as in the previous
lemma)

u(x) ≥ γ0
∫
Bx

v(y)qu(y)r

|x− y|n−α|y|σ1
dy

≥ C
∫
Bx

1

(1 + |y|)(n−α)(q+r)|x− y|n−α|y|σ1
dy

≥ C

(1 + |x|)(n−α)(q+r)

∫
Bx

1

|x− y|n−α|y|σ1
dy



6 E. CHERNYSH EJDE-2021/28

where, in this last step, we used that

(1 + |y|)(n−α)(q+r) ≤
(

1 +
3

2
|x|
)(n−α)(q+r)

≤
(3

2

)(n−α)(q+r)
(1 + |x|)(n−α)(q+r).

Thus, for |x| sufficiently large, we obtain the lower-bound estimate

u(x) ≥ C

(1 + |x|)(q+r)(n−α)+σ1

∫
Bx

1

|x− y|n−α
dy.

The estimate for u follows from the above once we observe that∫
Bx

1

|x− y|n−α
dy = C̃

∫ |x|/2
0

1

ρn−α
· ρn−1 dρ

= C̃

∫ |x|/2
0

ρα−1 dρ

= C̃|x|α ∼ C̃(1 + |x|)α.
This concludes the first step since a similar argument will yield the symmetric
inequality for v.

Step 2. Let (u, v) be a positive solution pair to (1.1). Then

u(x) & (1 + |x|)−(n−α) ln(1 + |x|), if (q + r)(n− α) = n− σ1,

v(x) & (1 + |x|)−(n−α) ln(1 + |x|), if (p+ s)(n− α) = n− σ2.
Proof of Step 2. We shall make use of an argument from Vétois [3] (see Theorem
1.1–Step 3.4 in this paper). An application of Lemma 2.2 shows that one shall
always have the estimates

u(x) & |x|α−n, v(x) & |x|α−n. (2.5)

For fixed k ∈ N, we define

A0 := inf
|x|<1

v(x), Ak := inf
2k−1<|x|<2k

v(x)

as well as

Ij,k := inf
2k−1<|x|<2k

∫
B(0,2j)\B(0,2j−1)

|x− y|α−n dy.

Let k ∈ N be large and fix x ∈ Rn such that 2k−1 < |x| < 2k. Using that
lim inf |x|→∞ ψ(x) > 0 we obtain for R > 0 and N ∈ N sufficiently large,

v(x) ≥ c
∫
|y|≥R

u(y)pv(y)s|x− y|α−n|y|−σ2 dy

≥ c
∑
j≥N

∫
B(0,2j)\B(0,2j−1)

u(y)pv(y)s|x− y|α−n|y|−σ2 dy.

Thus, by the estimates in (2.5),

v(x) ≥ c
∑
j≥N

∫
B(0,2j)\B(0,2j−1)

2−jp(n−α)−jσ2v(y)s|x− y|α−n dy

≥ c
∑
j≥N

∫
B(0,2j)\B(0,2j−1)

2−jp(n−α)−jσ2Asj |x− y|α−n dy

= c
∑
j≥N

2−jp(n−α)−jσ2Asj

∫
B(0,2j)\B(0,2j−1)

|x− y|α−n dy



EJDE-2021/28 ELLIPTIC SCHRÖDINGER SYSTEMS 7

≥ c
∑
j≥N

2−jp(n−α)−jσ2AsjIj,k.

This implies that there exists an N ∈ N and c > 0 such that for all positive integers
k sufficiently large

Ak ≥ c
∑
j≥N

2−j(p(n−α)+σ2)AsjIj,k. (2.6)

Now, let k be large and j ∈ {N,N + 1, . . . , k}; if 2k−1 < |x| < 2k we have∫
B(0,2j)\B(0,2j−1)

|x− y|α−n dy ≥ c2−k(n−α)
∫
B(0,2j)\B(0,2j−1)

dy

= c2−k(n−α) · (2nj − 2n(j−1))

which implies that for all k large,

Ij,k ≥ c2nj−k(n−α), ∀j ∈ {N,N + 1, . . . , k}. (2.7)

We now carry all we need in order to complete the proof. By (2.6)-(2.7), if k is an
integer much larger than N ,

Ak ≥ c
∑
j≥N

2−j(p(n−α)+σ2)AsjIj,k

≥ c
k∑

j=N

2−j(p(n−α)+σ2)AsjIj,k

≥ c
k∑

j=N

2−j(p(n−α)+σ2) · 2nj−k(n−α)Asj

≥ c2−k(n−α)
k∑

j=N

2−j(p(n−α)+σ2−n) · 2−sj(n−α)

= c2−k(n−α)
k∑

j=N

2−j((p+s)(n−α)+σ2−n)

= c2−k(n−α)(k −N).

Since k ∼ (k −N) as k →∞, it follows that

v(x) & |x|α−n ln |x|.

An identical argument applies to u in the case (q + r)(n − α) = n − σ1. This
concludes the proof of step 2.

The lower-bounds from the statement of the theorem follow immediately from
these previous two steps combined with Lemma 2.2. If u and v are assumed to be
weakly monotone decreasing, the upper-bounds follow from Proposition 2.1. �

3. Non-existence results

In this section we prove Theorem 1.4, which gives the non-existence results jus-
tifying our assumptions on the constants appearing in system (1.1). Throughout
this section, we assume that (1.2)-(1.3) hold and that both u and v are non-trivial.



8 E. CHERNYSH EJDE-2021/28

Lemma 3.1. Let f : Rn → (0,∞] be a weakly monotone decreasing function. Then

lim sup
|x|→∞

f(x) <∞.

Proof. Since f is weakly monotone decreasing we may take y ∈ R so large in
norm that f(x) ≤ Cf(y) whenever |x| ≥ |y|, where C is some positive constant
independent of x. Without loss of generality suppose that f(y) <∞. This implies
that lim sup|x|→∞ f(x) ≤ Cf(y) <∞, as was asserted. �

Proposition 3.2. System (1.1) does not admit any non-trivial weakly monotone
decreasing solution pairs when 0 < pq ≤ (1− r)(1− s).

Proof. For this proof we borrow ideas from Villavert [4, PROP-8] and Villavert [5,
THM-6]. Since we are handling the case pq > 0 we are assuming, especially, that
r, s ∈ [0, 1). We may also assume without loss of generality that σ1,2 ≥ 0. Suppose,
by way of contradiction, that (u, v) ∈ Wm(Rn)×Wm(Rn) is a positive solution pair
to system (1.1) when pq ≤ (1− r)(1− s). Using Lemma 2.2 it follows that

u(x) & |x|−b0

where we set b0 = n − α. Combining this with (1.3) shows that we may choose
R > 0 so large that u(x) ≥ c|x|−b0 , φ(x) ≥ γ0 > 0,

cu(x) ≤ u(y) and cv(x) ≤ v(y)

whenever |x| ≥ |y| ≥ R. For |x| sufficiently large we consider the annulus

Ax := {y ∈ Rn : R < |y| < |x|}.
Then

v(x) ≥ Cv(x)su(x)p|x|−σ2

∫
Ax

1

|x− y|n−α
dy ≥ Cv(x)su(x)p|x|α−σ2−n meas(Ax)

≥ Cv(x)su(x)p|x|α−σ2

≥ Cv(x)s|x|−pb0+α−σ2 , as |x| → ∞.
Hence,

v(x) ≥ C|x|−a1 where a1 :=
pb0 − α+ σ2

1− s
as |x| → ∞. Repeating this procedure and taking R sufficiently large in each step,
one can find by induction that

u(x) & |x|−bk and v(x) & |x|−ak

where

ak+1 :=
pbk − α+ σ2

1− s
and bk :=

qak − α+ σ1
1− r

.

The idea is to rewrite the induced recurrence relation in simpler terms to estimate
bk. Let us now define

P :=
p

1− s
, A :=

α

1− s
, Σ1 :=

σ1
1− r

,

Q :=
q

1− r
, B :=

α

1− r
, Σ2 :=

σ2
1− s

.

Using the above notation, we rewrite the recurrence relation of interest as

ak+1 := Pbk + Σ2 −A, bk := Qak + Σ1 −B.



EJDE-2021/28 ELLIPTIC SCHRÖDINGER SYSTEMS 9

By way of determining a closed form, let k ∈ N be large and 1 ≤ j ≤ k an integer.
The reader may verify by direct substitution that

bk = QjP jbk−j + (Q+Q2P +Q3P 2 + · · ·+QjP j−1)(Σ2 −A)

+ (1 +QP + · · ·+Qj−1P j−1)(Σ1 −B).

Now, taking j = k we find

bk = (PQ)kb0 +Q(Σ2 −A)

k−1∑
`=0

(PQ)` + (Σ1 −B)

k−1∑
`=0

(PQ)`. (3.1)

Which yields the following simple expression for bk,

bk = (PQ)kb0 + [Q(Σ2 −A) + (Σ1 −B)]

k−1∑
`=0

(PQ)`.

There are now two cases to distinguish.

Case 1: Assume pq = (1− s)(1− r). Then PQ = 1 so that bk → −∞ as k →∞.

Case 2: Suppose pq < (1− s)(1− r). We then have

0 < PQ =
pq

(1− s)(1− r)
< 1,

whence

bk = (PQ)kb0 + [Q(Σ2 −A) + (Σ1 −B)]
(PQ)k − 1

PQ− 1
.

Now, we calculate

Q(Σ2 −A) + (Σ1 −B) =
q

1− r
(
σ2 − α
1− s

) +
σ1 − α
1− r

=
q(σ2 − α) + (σ1 − α)(1− s)

(1− r)(1− s)
.

Finally,

PQ− 1 =
pq

(1− s)(1− r)
− 1 =

pq − (1− s)(1− r)
(1− s)(1− r)

,

whence

(Q(Σ2 −A) + (Σ1 −B))
1

PQ− 1
=
q(σ2 − α) + (σ1 − α)(1− s)

pq − (1− s)(1− r)
= −s0.

Under our conditions we have −s0 > 0 implying that bk < 0 for large enough k.
In either case we may make bk < 0 for all k ∈ N sufficiently large. Hence, for

suitable k it holds

u(x) & |x|−bk where bk < 0

which implies lim|x|→∞ u(x) =∞. However, this contradicts Lemma 3.1. �

Proposition 3.3. If p = 0 there is no positive solution pair to (1.1). Similarly,
there is no positive solution if q = 0.

Proof. Without loss of generality, we may assume that σ1,2 ≥ 0. We handle only
the case q = 0; a similar argument applies when p = 0. From Lemma 3.4 it follows
that u(x) ≥ c|x|−(n−α) as |x| → ∞, for some constant c > 0. Fix R > 0 so large



10 E. CHERNYSH EJDE-2021/28

that u(x) ≥ c|x|−(n−α) and φ(x) ≥ γ0 > 0 whenever |x| ≥ R (this can be done by
(1.3)). Given |x| ≥ 2R, we define as in the proof of Proposition 2.1

Ax :=
{
y ∈ Rn :

|x|
2
< |y| < |x|

}
so that

u(x) ≥
∫
Ax

φ(y)u(y)r

|x− y|n−α|y|σ1
dy

≥ c|x|−r(n−α)−σ1

∫
Ax

1

|x− y|n−α
dy

≥ c|x|−r(n−α)−σ1+α−n meas(Ax)

∼ c|x|−r(n−α)+α−σ1 , as |x| → ∞.
Or, rather,

u(x) & |x|−(rb0+σ1−α), where b0 := n− α.
Of course, we may repeat this argument inductively on k ∈ N to find that

u(x) & |x|−bk , where bk := rbk−1 + σ1 − α (3.2)

for all k ∈ N. By properties of a geometric sum, it is easy to verify that for each
k ∈ N

bk =

{
rkb0 + (σ1 − α) 1−rk

1−r , if r < 1,

b0 + k(σ1 − α), if r = 1.

Since σ1 < α, by taking k → ∞, we can make bk < 0 for some k ∈ N. Fix R > 0
large and assume that |x| < R; it then holds

u(x) ≥ c
∫
BR(0){

u(y)r

|x− y|n−α|y|σ1
dy

≥ c
∫
BR(0){

|y|−rbk+α−n−σ1 dy

≥ c
∫ ∞
R

ρ−rbk+α−σ1−1 dρ

where this last integral is convergent if and only if −rbk + α − σ1 < 0. Hence, we
obtain that u(x) =∞ in |x| < R: a contradiction. �

Having established these results, we must only show that the following holds.

Lemma 3.4. System (1.1) admits no positive solutions if either

−σ1 ≥ (q + r)(n− α)− α or − σ2 ≥ (p+ s)(n− α)− α.

Proof. We proceed by way of contradiction; without loss of generality assume that

−σ1 ≥ (q + r)(n− α).

By invoking Lemma 2.2, we may choose a constant C > 0 such that

u(x) ≥ C|x|−(n−α) and v(x) ≥ C|x|−(n−α)

for all |x| sufficiently large. Also, by (1.3), there exists γ0 > 0 such that φ(x) ≥ γ0
for all such x. Hence, for a sufficiently large R > 0 it holds

u(x) ≥ γ0
∫
|y|≥R

|y|−σ1
u(y)rv(y)q

|x− y|n−α
dy



EJDE-2021/28 ELLIPTIC SCHRÖDINGER SYSTEMS 11

≥ γ0
∫
|y|≥R

|y|(q+r)(n−α)−αu(y)rv(y)q

|x− y|n−α
dy

≥ Cγ0
∫
|y|≥R

|y|(q+r)(n−α)−(q+r)(n−α)−α

|x− y|n−α
dy

≥ C
∫
|y|≥R

|x− y|−n dy.

Since
∫
|y|≥R |x− y|

−n dy =∞, it follows that u ≡ ∞. This completes the proof of

the lemma. �

Proof of Theorem 1.4. Proposition 3.3 clearly implies that there does not exist a
positive solution if either q = 0 or p = 0. Likewise, it is a consequence of Proposition
3.2 that there does not exist any weakly monotone decreasing solutions whenever
pq ≤ (1− r)(1− s). The theorem then follows at once from Lemma 3.4. �

Acknowledgments. The author is very grateful to his supervisor, Professor Jérôme
Vétois, for his help and guidance as well as the opportunity to carry out research.
Many thanks to Dana Berman for valuable discussions.

References

[1] Li, Y.; Asymptotic behavior of positive solutions of equation ∆u + K(x)up = 0 in Rn. J.
Differential Equations 95 (1992) no. 2, 304-330.

[2] Liu, B.; Ma, L.; Symmetry results for decay solutions of elliptic systems in the whole space,
Adv. Math. 225 (2010), no. 6, 30523063.

[3] Vétois, Jérôme; Decay Estimates and Symmetry of Finite Energy Solutions to Elliptic Sys-

tems in Rn. Indiana University Mathematics Journal 68 (2019), no. 3, 663-696.
[4] Villavert, John; Qualitative properties of solutions for an integral system related to the Hardy-

Sobolev inequality. J. Differential Equations 258 (2015) no. 5, 1685-1714.

[5] Villavert, John; Sharp existence criteria for positive solutions of Hardy-Sobolev type systems.
Commun. Pure Appl. Anal. 14 (2) (2015) 493-515.

Edward Chernysh

Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada

Email address: edward.chernysh@mail.mcgill.ca


	1. Introduction and main results
	2. Decay estimates
	3. Non-existence results
	Acknowledgments

	References