Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 114, pp. 1–17.

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

ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS

TO ELLIPTIC EQUATIONS APPROACHING

CRITICAL GROWTH

ROSA PARDO, ARTURO SANJUÁN

Abstract. We study the asymptotic behavior of radially symmetric solutions
to the subcritical semilinear elliptic problem

−∆u = u
N+2
N−2 /[log(e+ u)]α in Ω = BR(0) ⊂ RN ,

u > 0, in Ω,

u = 0, on ∂Ω,

as α→ 0+. Using asymptotic estimates, we prove that there exists an explic-
itly defined constant L(N,R) > 0, only depending on N and R, such that

lim sup
α→0+

αuα(0)2

[log(e+ uα(0))]1+
α(N+2)

2

≤ L(N,R)

≤ 2∗ lim inf
α→0+

αuα(0)2

[log(e+ uα(0))]
α(N−4)

2

.

1. Introduction and main results

We consider the classical Dirichlet boundary value problem

−∆u = f(u) in Ω

u > 0 in Ω

u = 0 in ∂Ω

(1.1)

for u ∈ C2(Ω), in which Ω is an open bounded regular domain in RN , N > 2, and
f is locally-Lipschitz in [0,∞) and superlinear at infinity (i.e. lim inf f(u)/u > λ1

as u → ∞ where λ1 > 0 is the first eigenvalue of −∆ with Dirichlet boundary
conditions). We denote by 2∗ := 2N/(N−2) the critical Sobolev exponent. Namely,
H1(Ω) is compactly embedded in Lp(Ω) if and only if p < 2∗. The extended real
number f? := limu→∞ f(u)/u2∗−1 discriminates the problem (1.1) into three types:
critical if f? ∈ (0,∞), supercritical if f∗ =∞, and subcritical if f? = 0.

Pohozaev [15] discover that for the power nonlinearity f(u) = up with p ≥ 2∗−1,
there are no positive solutions to (1.1) in star-shaped domains. Bahri, Coron and

2010 Mathematics Subject Classification. 35B33, 35B45, 35B09, 35J60.
Key words and phrases. A priori bounds; positive solutions; semilinear elliptic equations;

Dirichlet boundary conditions; growth estimates; subcritical nonlinearites.
c©2020 Texas State University.

Submitted November 11, 2019. Published November 18, 2020.

1



2 [R. PARDO, A. SANJUÁN EJDE-2020/114

Ding show that (1.1) has a solution for some classes of non star-shaped domains,
see [3, 9]. The equivalence between uniform L2?(Ω) a-priori bounds and uniform
L∞(Ω) a-priori bounds in the subcritical case is proved in [4].

Assume that the nonlinearity is a pure subcritical power f(u) = u2∗−1−ε, ε > 0,
and Ω = BR (the open ball of radius R). Atkinson and Peletier [2] studied the
asymptotic behavior as ε→ 0+ of solutions to (1.1), and proved that

lim
ε→0+

εuε(0)2 = L(N,R),

and for all r 6= 0,

lim
ε→0+

uε(r)√
ε

= L̃(N,R)
( 1

rN−2
− 1

RN−2

)
.

Here L(N,R) and L̃(N,R) are constants only dependent on N , and R, defined by

L(N,R) :=
4

N − 2
[N(N − 2)]

N−2
2

Γ(N)

Γ(N/2)2

1

RN−2
, (1.2)

L̃(N,R) :=
(N − 2)

1
2

2
[N(N − 2)]

N−2
4

Γ(N/2)

Γ(N)1/2
R
N−2

2 =
[N(N − 2)]

N−2
2

L(N,R)1/2
, (1.3)

where Γ denotes the Gamma function. See also [11] with similar results for least
energy solutions on general domains.

We focus our attention on problem (1.1) with nonlinearity

f(u) = fα(u) :=
|u|2∗−2u

[log(e+ |u|)]α
. (1.4)

When α > 2
N−2 , there are a-priori L∞ bounds for classical positive solutions in

bounded, C2 domains, see [5, 6, 13, 14].
In [12], the existence of a-priori L∞ bounds for positive solutions is extended for

Hamiltonian elliptic systems −∆u = f(v),−∆v = g(u) with Dirichlet homogeneous
boundary conditions with

f(v) =
vp

[ln(e+ v)]α
, g(u) =

uq

[ln(e+ u)]β
,

1

p+ 1
+

1

q + 1
=
N − 2

N
,

and α, β > 2
N−2 .

Also for the p-Laplacian there are a-priori bounds for C1,µ(Ω) positive solutions
of elliptic equations −∆pu = f(u) with Dirichlet homogeneous boundary conditions
when

f(u) =
up

?−1

[ln(e+ u)]α
, p∗ =

Np

N − p
, α >

p

(N − p)
;

see [7]. This leads to a natural question: Is this lower bound on α a technical or
an intrinsic condition?

In this article we analyze the asymptotic behavior of solutions to

−∆u = u
N+2
N−2 /[log(e+ u)]α in Ω = BR(0) ⊂ RN ,

u > 0, in Ω,

u = 0, on ∂Ω,

(1.5)

as α → 0+. Firstly, we prove that for each α ∈
(
0, 2

N−2 ] fixed, the set of positive

solutions to (1.5) is a priori bounded. Henceforth, the bound from below on α in
[5, 6, 7, 12] are technical rather than intrinsic, at least when Ω is the open ball



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 3

of radius R. Secondly, we provide estimates for the growth of uα(0) and uα(r) as
α→ 0+. We adapt the techniques introduced by Atkinson and Peletier for the case
of subcritical powers in [1, 2].

Our first main result is on the existence of solutions to (1.5), and of L∞ a priori
bounds for each α > 0 fixed. The existence of solutions is already known due
to a result of Figueiredo, Lions and Nussbaum [8, Thm. 2.8] employing different
techniques involving elliptic regularity theory and topological variational methods.

Theorem 1.1. Fix α ∈
(
0, 2

N−2 ], let f = fα be as in (1.4) and assume Ω = BR.
Then the following results hold:

(i) There exists a radially symmetric solution to (1.5), u = uα(r) > 0.
(ii) There are constants A = Aα(N,R), B = Bα(N,R) > 0 depending only on

α, N and R, such that for every u = uα > 0, radially symmetric solution
to (1.5),

Aα(N,R) ≤ ‖uα‖L∞(Ω) ≤ Bα(N,R), for each α ∈
(
0,

2

N − 2

]
.

Our second main result is an estimate of the asymptotic behavior of uα(0) =
‖uα‖L∞(Ω) as α→ 0+.

Theorem 1.2. Let f = fα be as in (1.4) with α ∈
(

0, 2
N−2

]
, and Ω = BR. Then,

there exists a constant L(N,R) > 0 only depending on N and R (defined by (1.2)),
such that for any uα = uα(r), radially symmetric positive solution to (1.5), we have

lim sup
α→0+

αuα(0)2

[log(e+ uα(0))]1+
α(N+2)

2

≤ L(N,R), (1.6)

lim inf
α→0+

αuα(0)2

[log(e+ uα(0))]
α(N−4)

2

≥ 1

2∗
L(N,R). (1.7)

Our third main result is an estimate of the asymptotic behavior of uα(r) as
α→ 0+, when r 6= 0.

Theorem 1.3. Let fα(u) be as in (1.4) with α ∈
(
0, 2

N−2 ], and Ω = BR. Then,

there exists a constant L̃(N,R) > 0 only depending on N and R, such that for all
uα = uα(r), radially symmetric solution to (1.5) and for every r 6= 0, we have

lim inf
α→0+

[
[log(e+ uα(0))]

1
2 [1−α(N−6

2 )]uα(r)√
α

]
≥ L̃(N,R)

( 1

rN−2
− 1

RN−2

)
,

(1.8)

lim sup
α→0+

[
[log(e+ uα(0))]−α

N+4
4
uα(r)√
α

]
≤
√

2∗ L̃(N,R)
( 1

rN−2
− 1

RN−2

)
,

(1.9)

where L̃(N,R) is defined by (1.3).

In Section 2, keeping α ∈ (0, 2
N−2 ] and uα(0) = d > 0 fixed, we obtain lower

and upper estimate for radial solutions u = uα(r) of (1.5). In Section 3 we prove
Theorem 1.1 keeping α ∈ (0, 2

N−2 ] fixed, and allowing d to vary. In Section 4 we

prove Theorem 1.2 letting α→ 0+. Finally, in Section 5 we prove Theorem 1.3.



4 [R. PARDO, A. SANJUÁN EJDE-2020/114

2. Basic lemmas

In this Section we estimate uα(r) through several estimates of an auxiliary func-
tion, keeping α ∈

(
0, 2

N−2 ] and d > 0 fixed.

From Gidas, Ni and Nirenberg [10], it is well known that any positive solution
uα of (1.5) is radially symmetric and ∂uα

∂r < 0 for 0 < r < R. The search for radial
solutions of (1.5) leads to the ODE problem

u′′ +
N − 1

r
u′ + f(u) = 0 for r ∈ [0, R),

u(r) > 0 for r ∈ [0, R),

u(R) = 0, u′(0) = 0.

(2.1)

where, from now on f(u) = fα(u) is defined by (1.4). Let us consider the associated
initial-value problem

u′′ +
N − 1

r
u′ + f(u) = 0, for r > 0,

u(r) > 0,

u(0) = d, u′(0) = 0.

(2.2)

The Contraction Mapping Principle with parameters is applicable to (2.2) and for
each α ∈

(
0, 2

N−2 ] and d > 0 the initial-value problem (2.2) has a unique solution

u(r) = uα(r, d) depending continuously on α and d.
Since (2.2) is equivalent to(

rN−1 u′
)′

+ rN−1f
(
u(r)

)
= 0, 0 < r < R,

u(r) > 0,

u(0) = d, u′(0) = 0,

integrating on [0, r] we have

rN−1u′(r) = −
∫ r

0

sN−1f
(
u(s)

)
ds < 0,

and the solutions are decreasing. It is clear that there exist solution to (2.1) if there
exists some d such that uα(R, d) = 0. Set

t :=
(N − 2

r

)N−2

, y(t) := u(r),
(
y(t) = yα(t, d) = uα(r, d)

)
, (2.3)

problem (2.2) becomes the backward problem

y′′ + t−
2(N−1)
N−2 f(y(t)) = 0 for t <∞,

y(t) > 0,

lim
t→+∞

y(t) = d, lim
t→+∞

y′(t) = 0.

(2.4)

When the nonlinearity is f(s) = Asp, for some A > 0, equation (2.4) is known as
the Emden-Fowler equation.

Integrating y′′ on (t,+∞), see (2.4),

y′(t) =

∫ ∞
t

s−
2(N−1)
N−2 f

(
y(s)

)
ds (2.5)



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 5

Integrating now y′ on (t,+∞), and from Fubini’s Theorem

y(t) = d−
∫ ∞
t

(s− t)s−
2(N−1)
N−2 f(y(s)) ds. (2.6)

Throughout this section we keep α ∈
(
0, 2

N−2 ] and d > 0 fixed. Define

T (d) = Tα(d) := inf{t > 0 : y(t) > 0}. (2.7)

By definition T (d) ≥ 0, and since continuous dependence on the parameters, T (d)
is continuous. We will prove in Lemma 2.4 that T (d) > 0, therefore we can define

R(d) := (N − 2)/T (d)
1

N−2 . Obviously, u = uα(r, d) is a solution to (2.1) on (0, R)
if and only if for each α ∈

(
0, 2

N−2 ], there exists some d > 0 (depending on α), such

that R(d) = R, or in other words,

T (d) :=
(N − 2

R

)N−2

. (2.8)

Let

Dα := {d = dα > 0 : Tα(d) = [(N − 2)/R]N−2}. (2.9)

By [8, Thm 2.8], problem (2.1) has a solution. In other words, Dα 6= ∅. Our first
aim is to prove that, for α fixed, the set Dα is bounded. We denote

z(t) = zα(t, d) := dt
[
t

2
N−2 +

(N − 2)f(d)

Nd

]−N−2
2

. (2.10)

By direct computations we can show that z satisfies the Emden-Fowler equation

z′′(t) + t−
2(N−1)
N−2

1

[log(e+ d)]α
z(t)2∗−1 = 0, for t > 0

z(t) > 0

z(0) = 0, lim
t→+∞

z(t) = d, lim
t→+∞

z′(t) = 0.

(2.11)

Obviously z′′ < 0, and integrating z′′ on (t,+∞), then z′ > 0. Moreover, in its
integral form, (2.11) is equivalent to

z(t) = d− 1

[log(e+ d)]α

∫ ∞
t

(s− t)s−
2(N−1)
N−2 z(s)2∗−1 ds. (2.12)

The function z will be useful in estimating y. For instance we have the following
result proved in [1, Lemma 1.(iii) and Remark 1].

Lemma 2.1. Fix α ∈
(
0, 2

N−2 ] and d > 0. Let y = y(t, d) solve (2.4), and

z = z(t, d) solve (2.11). Then

y(t, d) < z(t, d) for every t > T (d).

Using (2.11) it is easy to see that for t ≥ 0, the function z is increasing and
concave. Then for every t > 0, z(t) < min{z′(0)t, d}. A direct computation using
(2.10) shows that z′(0) = N1M(d) where

N1 :=
( N

N − 2

)N−2
2

, and M = M(d) :=
log(e+ d)

α(N−2)
2

d
. (2.13)

Hence, we have the following consequence of Lemma 2.1.



6 [R. PARDO, A. SANJUÁN EJDE-2020/114

Lemma 2.2. Fix α ∈
(
0, 2

N−2 ] and d > 0. Let y = y(t, d) solve (2.4). Then

y(t) < min{N1M(d)t, d} for every t > T (d), (2.14)

where N1, and M(d) are defined by (2.13)

For further estimates we introduce the Pohozaev functional

H(t) :=
1

2
t
(
y′(t)

)2 − 1

2
y(t)y′(t) +

(1

t

) N
N−2F

(
y(t)

)
, for t ≥ T (d), (2.15)

where F (s) =
∫ s

0
f(t) dt. The following lemma states some properties of H.

Lemma 2.3. Fix α ∈
(
0, 2

N−2 ] and d > 0. Let y = y(t, d) solve (2.4). Then

the Pohozaev functional (2.15) satisfies H ′(t) < 0 for t > T (d) and H(t) ↘ 0 as
t→∞. In particular H(t) > 0 for t ≥ T (d).

Proof. Integrating F (t) by parts,

F (t) =
1

2∗

[
tf(t) + α

∫ t

0

s2∗

[log(e+ s)]α+1(e+ s)
ds
]
. (2.16)

Differentiating (2.15) and using (2.4), we have

H ′(t) = −α
2

(1

t

) 2(N−1)
N−2

∫ y(t)

0

s2∗

[log(e+ s)]α+1(e+ s)
ds < 0, (2.17)

which proves the first claim of the lemma.
Substituting (2.16) in (2.15), we obtain

H(t) =
1

2
t(y′)2 − 1

2
yy′ +

1

2∗
(1

t

) N
N−2

[
yf(y) (2.18)

+ α

∫ y(t)

0

s2∗

[log(e+ s)]α+1(e+ s)
ds
]
. (2.19)

By L’Hopital’s Rule and (2.4),

lim
t→∞

ty′(t) = lim
t→∞

(1

t

) 2
N−2 f

(
y(t)

)
= 0, (2.20)

hence t(y′)2 → 0 as t → ∞. Therefore, the first term in the right hand side of
(2.18) tends to 0 as t→∞. Since the asymptotic behavior of y, and y′ as t→∞.
The second, third and fourth terms in the right hand side of (2.18) also tend to 0
as t→∞. Then H(t)→ 0 as t→∞.

Since H ′ < 0, H(t) ↘ 0 as t → ∞, consequently H(t) > 0 for t ≥ T (d). This
completes the proof. �

The above lemmas are useful for proving the positiveness of T (d).

Lemma 2.4. Fix α ∈
(
0, 2

N−2 ]. Let T = T (d) be defined by (2.7). Then

T (d) > 0, for every d > 0.

Proof. Assume by contradiction that T (d) = 0. From Lemma 2.3, H(0) > 0.

Moreover, from F (s) =
∫ s

0
f(t) dt ≤ s2

∗

2∗ , and Lemmas 2.2 and 2.3, we have

t−( N
N−2 )F (y(t)) ≤ 1

2∗
t−( N

N−2 )y(t)2∗ ≤ 1

2∗
(
N1M(d)

)2∗
t
N
N−2 → 0 as t→ 0+.

This and (2.15) imply that H(0) = − 1
2y(0)y′(0) = 0, contradicting Lemma 2.3. �



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 7

We now look for a lower estimate for y. Let

T̃ (d) = T̃α(d) :=
d2

log(e+ d)
α(N−2)

2

, (2.21)

then, for every ε > 0,

z
(
εT̃ (d)

)
= cεd, with cε :=

ε

[N−2
N + ε

2
N−2 ]

N−2
2

. (2.22)

Observe that
cε
ε
→ N1 as ε→ 0, and T̃ (d) =

d

M(d)
, (2.23)

see (2.13). Next, we state a lower bound of y.

Lemma 2.5. Let y = y(t, d) solve (2.4), and z = z(t, d) solve (2.11). For every
ε > 0, there exists d0 = d0(ε) and some c′ε,d > 0 for d ≥ d0, such that

y(t) >
[
1− α

(3

2

)α
c′ε,d
]
z(t) for every t > εT̃ (d).

Proof. Fix any ε > 0, and any d > 0. Take t > εT̃ (d). Since (z > y and f ↗), from
(2.12), using the Mean Value Theorem with θ ∈ (z, d), with θ > z > cεd, using
(2.12), and d < z/cε, we deduce that

y(t) > d−
∫ ∞
t

(s− t)s−
2(N−1)
N−2 f(z) ds

= z −
∫ ∞
t

(s− t)s−
2(N−1)
N−2 z2∗−1

[ 1

[log(e+ z)]α
− 1

[log(e+ d)]α

]
ds

= z − α
∫ ∞
t

(s− t)s−
2(N−1)
N−2 z2∗−1 d− z

[log(e+ θ)]α+1(θ + e)
ds

≥ z − αd

[log(e+ cεd)]α+1(cεd+ e)

∫ ∞
t

(s− t)s−
2(N−1)
N−2 z2∗−1 ds

≥ z − α

cε[log(e+ cεd)]α+1

∫ ∞
t

(s− t)s−
2(N−1)
N−2 z2∗−1 ds

= z − α[log(e+ d)]α

cε[log(e+ cεd)]α+1
(d− z)

≥ z
[
1− α (1− cε)

c2ε

[log(e+ d)]α

[log(e+ cεd)]α+1

]
.

Consequently, for all ε > 0, and d > 0 fixed,

y(t) ≥
[
1− α (1− cε)

c2ε

[log(e+ d)]α

[log(e+ cεd)]α+1

]
z(t), for any t > εT̃ (d). (2.24)

Let us keep ε > 0 fixed and allow d to be large. Since log(d+e)
log(e+cεd) → 1 as d→∞,

there exists d0 = d0(ε) such that log(d+e)
log(e+cεd) < 3/2, for all d ≥ d0, in fact we can

define

d0 = d0(ε) :=
1

c3ε
,

where cε is defined by (2.22). Now, taking

c′ε,d :=
1− cε
c2ε

1

log(e+ cεd)
, (2.25)



8 [R. PARDO, A. SANJUÁN EJDE-2020/114

the proof is complete. �

Lemma 2.6. Let y = yα(t, d) solve (2.4), and z = zα(t, d) solve (2.11). For every
ε > 0, there exists d1 = d1(ε), such that for all d ≥ d1

y(t) > γ(α)z(t) for every t > εT̃ (d). (2.26)

where

γ(α) := 1− N − 2

4
α ≥ 1

2
, for all α ∈

(
0,

2

N − 2

]
. (2.27)

In particular, for every ε ∈
(
0,
(

2
N

)N−2
2
)
,

y
(
εT̃ (d)

)
≥ 1

2
εd, for all d ≥ d1.

Proof. For ε > 0 fixed, let us define

d1 = d1(ε) :=
1

cε
exp

[ 4

N − 2
(
3

2
)α

1− cε
c2ε

]
, (2.28)

where cε is given by (2.22). Hence,

1− α
(3

2

)α
c′ε,d ≥ 1− N − 2

4
α ≥ 1

2
, for all d ≥ d1, and α ∈

(
0,

2

N − 2

]
,

which, combined with Lemma 2.5, proves (2.26).
In particular, for ε ∈ (0, ε0), and d ≥ d1(ε),

y
(
εT̃ (d)

)
≥ 1

2
z
(
εT̃ (d)

)
=

1

2

εd

[N−2
N + ε

2
N−2 ]

N−2
2

≥ 1

2

εd

[N−2
N + ε

2
N−2

0 ]
N−2

2

,

choosing ε0 := ( 2
N )

N−2
2 we obtain

y
(
εT̃ (d)

)
≥ 1

2
εd > 0,

which compltes the proof. �

3. Further estimates and proof of Theorem 1.1

In this Section we estimate u = uα(r, d) through several estimates of the auxiliary
function y = yα(t, d) and in particular of T = Tα(d), keeping α ∈ (0, 2

N−2 ] fixed
and allowing d to vary. As an immediate consequence of Lemmas 2.5-2.6 we have
the following lemma.

Lemma 3.1. Let T̃ (d) be defined by (2.21). Then

T (d) = o(T̃ (d)) as d→∞. (3.1)

Proof. Lemma 2.6 state in particular that for any ε > 0 small enough, there exists
d1 = d1(ε), such that for all d ≥ d1,

y
(
εT̃ (d)

)
≥ 1

2
εd > 0.

Therefore, from definition of T (d), for any ε > 0, and d ≥ d1(ε), T (d) < εT̃ (d). �



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 9

Now, we introduce the Hardy asymptotic notation. For f, g : R → R+, we say
that

f(d) . g(d) as d→ d0, with 0 ≤ d0 ≤ ∞, if lim sup
d→d0

|f(d)|
|g(d)|

< +∞.

In a similar way we use the notation f(d) & g(d) as d→ d0, if lim supd→d0
|g(d)|
|f(d)| <

+∞. Finally we will use the notation

f(d) = Θ(g(d)) as d→ d0, with 0 ≤ d0 ≤ ∞,
to denote f . g and g . f as d → d0. The following lemma relate to estimations
of y(t) and y′(t) for specific values of t when d is large.

Lemma 3.2. Let y = y(t, d) solve (2.4). Let T = T (d), T̃ = T̃ (d) and M = M(d)
be defined by (2.7), (2.21) and (2.13) respectively. Then, the following holds:

(i) y(2T ) = o(d), as d→∞.
(ii) There exists a constant CN,α depending only on N and α, explicitly defined

by (3.2), such that

y(T̃ (d)) ≥ CN,α d, as d→∞.
(iii) y′(2T ) = Θ

(
M(d)

)
, as d→∞.

(iv) y(t, d) = Θ
(
M(d)

(
t− T (d)

))
, as d→∞, uniformly for every t ∈ [2T, T̃ ].

Proof. (i) Using (2.14) with t = 2T (d), (2.21)-(2.23) and (3.1), we obtain

y(2T )

d
≤ 2N1

T (d)

T̃ (d)
→ 0 as d→ +∞.

(ii) Taking ε = 1 in Lemma 2.6, and from (2.22), we can write

y(T̃ (d)) ≥
(

1− N − 2

4
α
)
z(T̃ (d)) ≥ CN,α d,

where

CN,α :=
(

1− N − 2

4
α
)( N

2(N − 1)

)N−2
2

. (3.2)

(iii) Using that y′′ < 0, Lemma 2.1, (2.10), and Lemma 3.1, we deduce

y′(2T ) <
y(2T )− y(T )

T
≤ z(2T )

T

=
2d

[(2T )
2

N−2 + fα(d)

( N
N−2 )d

]
N−2

2

≤ 2d
( N

N − 2

)N−2
2 [log(e+ d)]α

N−2
2

d2

≤ 2N1M(d).

On the other hand, using again y′′ < 0, (i), (ii), and Lemma 3.1 we obtain

y′(2T ) >
y(T̃ (d))− y(2T )

T̃ (d)− 2T
≥ CN,αd− y(2T )

T̃ (d)− 2T
≥ CN,α − ε

1 + ε
M(d) ≥ 1

2
CN,αM(d).

(iv) Since y′′ < 0, y(T ) = 0, and Lemma 2.2, it follows that

y(t, d)

t− T (d)
≤ y(2T )

T (d)
.M(d)



10 [R. PARDO, A. SANJUÁN EJDE-2020/114

uniformly with respect to t ∈ [2T, T̃ ]. On the other hand, using y′′ < 0, (ii),
Lemma 3.1, and (2.23)

y(t, d)

t− T (d)
≥ y(T̃ )

T̃ − T
&

d

T̃ (d)
= M(d),

uniformly with respect to t ∈ [2T, T̃ ]. This completes the proof. �

To prove the lower and upper bounds in Theorem 1.1 we need the following two
lemmas.

Lemma 3.3. Let T = T (d) be defined by (2.7). Then

0 < T (d) ≤
(N − 2

2∗

)N−2
2 d2

[log(e+ d)]
α(N−2)

2

, (3.3)

and in particular
T (d) . d2 as d→ 0+.

Proof. Since (2.6), Lemma 2.2, and f is increasing, it follows that

0 = y(T ) ≥ d− f(d)

∫ ∞
T

(s− T )s−
2(N−1)
N−2 ds = d− f(d)

2∗

N−2

( 1

T

) 2
N−2 ,

then (3.3) holds. We complete the proof by letting d→ 0. �

Lemma 3.4. Let T = T (d) be defined by (2.7) and keep α ∈
(

0, 2
N−2 ] fixed. Then

T (d) &
d2

[log(e+ d)]
α(N−2)

2 +1
, as d→∞.

Proof. From Lemmas 2.3 and 3.1, it is clear that

H(2T ) > H(2T )−H(T̃ ) =

∫ T̃

2T

(−H ′(s)) ds >
∫ T̃

T̃ /2

(−H ′(s)) ds.

By L’Hopital’s Rule, it is easy to prove that for m > 1 and β > 0,

lim
t→∞

∫ t
0

sm−1

[log(e+s)]β
ds

tm

log(t+e)β
=

1

m
, lim

t→∞

∫ t
0

sm

[log(e+s)]β(s+e)
ds

tm

log(t+e)β
=

1

m
. (3.4)

Therefore,
F (t) = Θ(tf(t)), as t→∞. (3.5)

We notice that

M(d)(s− T (d)) = Θ(d) uniformly for s ∈ [T̃ /2, T̃ ], (3.6)

see (2.23) and Lemma 3.1. Now using (2.17) and part (iv) of Lemma 3.2 we deduce
the following:

H(2T ) &
∫ T̃

T̃ /2

s−
2(N−1)
N−2

y(s)2∗

[log(e+ y(s))]α+1
ds (by (3.4) and (3.6))

&
∫ T̃

T̃ /2

s−
2(N−1)
N−2

(
M(d)(s− T )

)2∗
[log(e+M(d)(s− T ))]α+1

ds (by Lemma 3.2 (iv))

&
[log(e+ d)]αN−α−1

d2∗

∫ T̃

T̃ /2

s−
2(N−1)
N−2 (s− T )2∗ ds (using (3.6))



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 11

&
[log(e+ d)]αN−α−1

d2∗

(
T̃
) N
N−2 (by Lemma 3.1)

= [log(e+ d)]
α(N−2)

2 −1.

Note that α(N−2)
2 − 1 ≤ 0. Using Lemma 3.2 (iii) and (iv), we have

H(2T ) < Ty′(2T )2 + (2T )−( N
N−2 )F

(
y(2T )

)
(by (2.15))

. T
(
M(d)

)2
+

1

T
N
N−2

y(2T )2∗

[log(e+ y(2T ))]α
(by Lemma 3.2 (iii) and (3.5))

. T
(
M(d)

)2
+

M(d)2∗T
N
N−2

[log(e+M(d)T )]α
(by Lemma 3.2 (iv))

(3.7)

Denoting S(d) := T (d)
(
M(d)

)2
, we can write

5H(2T ) . S(d) + S(d)
N
N−2 [log(e+ S(d)/M(d))]−α.

From Lemma 3.1 we know that S(d) = o
(

[log(e+ d)]
α(N−2)

2

)
, and from Lemma 3.4

that

S(d) & [log(e+ d)]
α(N−2)

2 −1, as d→∞.

Hence S(d)
M(d) &

d
log(e+d) . Moreover, since log

(
e+ d

log(e+d)

)
= Θ

(
log(e + d)

)
as

d→∞, we have[
log
(
e+

S(d)

M(d)

)]−α
. [log(e+ d)]−α, and

S(d)
2

N−2

[log(e+ S(d)/M(d))]α
= o(1).

Consequently H(2T ) . S(d) and

T (d) &
d2

[log(e+ d)]
α(N−2)

2 +1
.

�

Proof of Theorem 1.1. (i) Fix α ∈ (0, 2
N−2 ]. From Lemmas 3.3 and3.4, and the

continuity of T (d), there exists a d = dα ∈ (0,∞) such that T (dα) = [(N −
2)/R]N−2. The corresponding solutions of the IVP (2.2) is a radial solution of the
BVP (2.1).

(ii) Fix α ∈
(

0, 2
N−2

]
. Assume on the contrary that there exists a sequence of

solutions to (2.1), denoted by un, such that dn := un(0) = ‖un‖∞ → 0 as n→∞.
By Lemma 3.3, Tn := T (dn) → 0 as dn → 0+. But un = uα,n is a solution to
(2.1), and therefore yn := yα,n is a solution to (2.4) with T (dn) = [(N − 2)/R]N−2

constant, contradicting that T (dn)→ 0 as dn → 0+. Therefore, there is a constant
A > 0 such that A < ‖u‖∞.

On the other hand, assume on the contrary that there exists a sequence of
solutions to (2.1), denoted by un, such that dn := un(0) = ‖un‖∞ →∞ as n→∞.
By Lemma 3.4, T (dn) → ∞ as dn → ∞. But reasoning as before, T (dn) =
[(N − 2)/R]N−2, a constant value, contradicting that T (dn) → ∞ as dn → ∞.
Therefore, there exists a constant B > 0 such that ‖u‖∞ < B. This completes the
proof. �



12 [R. PARDO, A. SANJUÁN EJDE-2020/114

4. Proof of Theorem 1.2

In this Section, we consider only values of d = dα ∈ Dα, where Dα is defined by
(2.9), and allow α to vary. As a consequence T = Tα(d) is fixed and defined by

T = Tα(d) =
(N − 2

R

)N−2

, ∀d = dα ∈ Dα, ∀α ∈
(

0,
2

N − 2

]
,

and uα(r, d) is a solution of (2.1) for d ∈ Dα.

Lemma 4.1. Let Dα be defined by (2.9). Then

lim
α→0+

inf Dα = +∞.

Proof. Assume by contradiction that there is a sequence αn ↘ 0 and some M0 > 0
such that inf Dαn < M0. Then, there is a subsequence dn ∈ Dαn such that dn < M0

for every n. Hence, there is an ε0 > 0 depending only on M0, such that

ε0T̃ (dn) = ε0
d2
n

[log(e+ dn)]
α(N−2)

2

≤
(N − 2

R

)N−2
= T, for every n.

Then, firstly from (2.24), and secondly from dn < M0, there is an α0 > 0 such that
for every αn ∈ (0, α0),

0 = y(T, dn) >
[
1− αn

1− cε0
c2ε0

[log(e+ dn)]αn

[log(e+ cε0dn]αn+1

]
z(T, dn) > 0,

which is a contradiction. �

To obtain new estimates, we will use the incomplete beta function defined as

B(x, a, b) =

∫ ∞
x

ta−1(1 + t)−a−b dt, a, b > 0.

In [2, Lemma A2] a slightly variant of the following relation is proved∫ ∞
t

s−
2(N−1)
N−2 zr(s, d) ds

= N1
N

2
dr−2∗ [log(e+ dα)]α

N
2 B
((N1t

T̃

) 2
N−2

,
r − N

N−2
2

N−2

,
N

2

)
,

(4.1)

with r > N
N−2 . We denote

I(α) :=
T
(
y′α(T )

)2
α

=

∫ ∞
T

t−
2(N−1)
N−2

(∫ yα(t)

0

s2∗

[log(e+ s)]α+1(e+ s)
ds
)
dt. (4.2)

This equality is a consequence of (2.17) and (2.15).

Lemma 4.2. Let y = yα(t, d) solve (2.4), and let Dα and Iα be defined by (2.9)
and (4.2) respectively. Then

(i)

lim sup
α→0+

sup
dα∈Dα

[ d2
α

[log(e+ dα)]αN
T
(
y′α(T )

)2] ≤ N2
1T.

(ii)

lim inf
α→0+

inf
dα∈Dα

[ d2
α

[log(e+ dα)]α(N−2)
T
(
y′α(T )

)2] ≥ N2
1T. (4.3)



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 13

(iii)

lim sup
α→0+

sup
dα∈Dα

[ 1

[log(e+ dα)]α
N
2

I(α)
]
≤ N1

N

2

Γ(N2 )2

Γ(N)
. (4.4)

(iv)

lim inf
α→0+

inf
dα∈Dα

[
[log(e+ dα)]1−

α(N−2)
2 I(α)

]
≥ N1

N − 2

4

Γ(N2 )2

Γ(N)
. (4.5)

Proof. (i) From (2.5), Lemma 2.1, and (4.1) with t = T and r = 2∗ − 1, we have

y′α(T ) =

∫ ∞
T

t−
2(N−1)
N−2 f

(
yα(t)

)
dt

≤
∫ ∞
T

t−
2(N−1)
N−2 f

(
zα(t)

)
dt

≤
∫ ∞
T

t−
2(N−1)
N−2 zα(t)2∗−1 dt

= N1
N

2

[log(e+ dα)]α
N
2

dα
B
((TN1

T̃

) 2
N−2

, 1,
N

2

)
≤ N1

N

2

[log(e+ dα)]α
N
2

dα
B
(
0, 1,

N

2

)
= N1

[log(e+ dα)]α
N
2

dα
.

(4.6)

Hence

lim sup
α→0+

sup
dα∈Dα

[ dα

[log(e+ dα)]α
N
2

y′α(T )
]
≤ N1, (4.7)

which proves part (i).

(ii) Fix an arbitrary ε > 0. From (2.5), Lemma 3.1, Lemma 2.6 and (4.1), there
exists a d1 only depending on ε (see (2.28)), such that for every dα ≥ d1

y′α(T ) >

∫ ∞
εT̃

s−
2(N−1)
N−2

yα(s)2∗−1

[log(e+ yα(s)]α
ds

>
γ(α)2∗−1

[log(e+ dα)]α

∫ ∞
εT̃

s−
2(N−1)
N−2 zα(s)2∗−1 ds

= N1
N

2

γ(α)2∗−1

[log(e+ dα)]α
[log(e+ dα)]α

N
2

dα
B
(

(εN1)
2

N−2 , 1,
N

2

)
.

(4.8)

The inequality dα ≥ d1 for α small enough, holds thanks to Lemma 4.1. Hence

inf
dα∈Dα

[ dα

[log(e+ dα)]
α(N−2)

2

y′α(T )
]
≥ N1

N

2
γ(α)2∗−1B

(
(εN1)

2
N−2 , 1,

N

2

)
, (4.9)

for an arbitrary ε > 0 fixed. Because γ(α) → 1 as α → 0+, see (2.27), and
by continuity of the incomplete beta function with respect to its first argument,

B
(
(εN1)

2
N−2 , 1, N2

)
→ B

(
0, 1, N2

)
as ε→ 0. Therefore,

lim inf
α→0+

inf
dα∈Dα

[ dα

[log(e+ dα)]
α(N−2)

2

y′α(T )
]
≥ N1. (4.10)

part (ii) has been proved.



14 [R. PARDO, A. SANJUÁN EJDE-2020/114

(iii) Since the integrand in (4.2) is increasing, by Lemma 2.1 and (4.1), we have

I(α) =

∫ ∞
T

t−
2(N−1)
N−2

(∫ yα(t)

0

s2∗

[log(e+ s)]α+1(e+ s)
ds
)
dt

≤
∫ ∞
T

t−
2(N−1)
N−2

yα(t)2∗+1

[log(e+ yα(t)]α+1
(
e+ yα(t)

) dt
≤
∫ ∞
T

t−
2(N−1)
N−2 yα(t)2∗ dt

<

∫ ∞
T

t−
2(N−1)
N−2 zα(t)2∗ dt

= N1
N

2
[log(e+ dα)]α

N
2 B
((TN1

T̃

) 2
N−2

,
N

2
,
N

2

)
.

(4.11)

Hence

lim sup
α→0+

sup
dα∈Dα

I(α)

[log(e+ dα)]α
N
2

≤ N1
N

2
B
(

0,
N

2
,
N

2

)
= N1

N

2

Γ(N2 )2

Γ(N)
. (4.12)

This proves part (iii).

(iv) Fix an arbitrary ε > 0 and δ ∈ (0, 1). From (4.2), Lemma 3.1, (3.4), Lemma
2.6 and (4.1), there exists a d1 only depending on ε (see (2.28)), such that for every
dα ≥ d1,

I(α) =

∫ ∞
T

t−
2(N−1)
N−2

(∫ yα(t)

0

s2∗

[log(e+ s)]α+1(e+ s)
ds
)
dt

≥
∫ ∞
εT̃

t−
2(N−1)
N−2

(∫ yα(t)

0

s2∗

[log(e+ s)]α+1(e+ s)
ds
)
dt

≥ 1− δ
2∗

∫ ∞
εT̃

t−
2(N−1)
N−2 yα(t)2∗

[log(e+ yα(t)]α+1
dt

≥ (1− δ)γ(α)2∗

2∗[log(e+ d)]α+1

∫ ∞
εT̃

t−
2(N−1)
N−2 zα(t)2∗ dt

= N1
N

2

(1− δ)γ(α)2∗

2∗
[log(e+ dα]α(N−2

2 )−1B
(

(εN1)
2

N−2 ,
N

2
,
N

2

)
.

(4.13)

Since γ(α)→ 1 as α→ 0+, see (2.27), it follows that

inf
dα∈Dα

[
[log(e+ dα)]1−

α(N−2)
2 I(α)

]
≥ N − 2

4
N1(1− δ)B

(
(εN1)

2
N−2 ,

N

2
,
N

2

)
,

for an arbitrary ε > 0 fixed. Again, by the continuity of the incomplete beta

function with respect to its first argument, B
(
(εN1)

2
N−2 , 1, N2

)
→ B

(
0, 1, N2

)
as

ε→ 0, and

lim inf
α→0+

inf
dα∈Dα

[
[log(e+ dα)]1−

α(N−2)
2 I(α)

]
≥ N − 2

4
N1(1− δ)B

(
0,
N

2
,
N

2

)
=
N − 2

4
N1(1− δ)

Γ(N2 )2

Γ(N)
.

for δ ∈ (0, 1) arbitrary, this completes the proof of (iv) and of the Lemma. �



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 15

Proof of Theorem 1.2. Recall that uα(0) = dα. Using (4.2), Lemma 4.2 (i) and
(4.5), and from definition of T , see (2.8), we have

lim sup
α→0+

( αuα(0)2

[log(e+ uα(0))]1+
α(N+2)

2

)
= lim sup

α→0+

sup
dα∈Dα

( αd2
α

[log(e+ dα)]1+
α(N+2)

2

)
≤ lim sup

α→0+

sup
dα∈Dα

( d2
αTy

′
α(T )2

[log(e+ dα)]αN

)
lim sup
α→0+

sup
dα∈Dα

( [log(e+ dα)]−1+
α(N−2)

2

I(α)

)
≤ 2

N
N1 2∗

Γ(N)

Γ(N/2)2
T

=
4

N − 2
[N(N − 2)](N−2)/2 Γ(N)

Γ(N/2)2

1

RN−2
= L(N,R),

and (1.6) has been proved.
Now we prove (1.7). Using (4.2), Lemma 4.2, (4.3) and (4.4) we have

lim inf
α→0+

( αuα(0)2

[log(e+ uα(0))]α(N−4)/2

)
= lim inf

α→0+
inf

dα∈Dα

( αd2
α

[log(e+ dα)]α(N−4)/2

)
≥ lim inf

α→0+
inf

dα∈Dα

( d2
αTy

′
α(T )2

[log(e+ dα)]α(N−2)

)
lim inf
α→0+

inf
dα∈Dα

( [log(e+ dα)]α
N
2

I(α)

)
≥ 2

N
N1

Γ(N)

Γ(N2 )2
T

=
2

N
[N(N − 2)]

N−2
2

Γ(N)

Γ(N/2)2

1

RN−2
=

1

2∗
L(N,R).

(4.14)

Assertion (1.7) has been proved. This completes the proof of Theorem 1.2. �

5. Proof of Theorem 1.3

Theorem 1.3 will be a consequence of Lemma 4.2 and the following lemma.

Lemma 5.1. Let y = yα(t, d) solve (2.4), and let Dα be defined by (2.9). Then,
the following estimates hold

(i) For every t ≥ T ,

lim sup
α→0+

sup
dα∈Dα

[ dα

[log(e+ dα)]α
N
2

yα(t)
]
≤ N1(t− T ). (5.1)

(ii)

lim inf
α→0+

inf
dα∈Dα

[ dα

[log(e+ dα)]
α(N−2)

2

yα(t)
]
≥ N1(t− T ). (5.2)

Proof. (i) Using the concavity of y, we deduce y′α(t) ≤ y′α(T ) for every t ≥ T . Now,
integrating (4.6) we obtain (5.1).

(ii) Fix an arbitrary ε > 0. Let us take t ∈ (T, εT̃ ). Since concavity of y, from
(2.5), Lemma 2.6 and (4.1), there exists a d1 only depending on ε, see (2.28), such



16 [R. PARDO, A. SANJUÁN EJDE-2020/114

that for every dα ≥ d1,

y′α(t) ≥ y′α
(
εT̃ (d)

)
=

∫ ∞
εT̃

s−
2(N−1)
N−2

yα(s)2∗−1

[log(e+ yα(s)]α
ds

>
γ(α)2∗−1

[log(e+ dα)]α

∫ ∞
εT̃

s−
2(N−1)
N−2 zα(s)2∗−1 ds

= N1
N

2

γ(α)2∗−1

[log(e+ dα)]α
[log(e+ dα)]α

N
2

dα
B
(

(N1ε)
2

N−2 , 1,
N

2

)
.

Then

inf
dα∈Dα

[ dα

[log(e+ dα)]
α(N−2)

2

y′α(t)
]
≥ N1

N

2
γ(α)2∗−1B

(
(N1ε)

2
N−2 , 1,

N

2

)
.

Since γ(α)→ 1 as α→ 0+, for every t > T ,

lim inf
α→0+

inf
dα∈Dα

[ dα

[log(e+ dα)]
α(N−2)

2

y′α(t)
]
≥ N1

N

2
B
(

(N1ε)
2

N−2 , 1,
N

2

)
,

for an arbitrary ε > 0 fixed. By continuity of the incomplete beta function with

respect to its first argument, B
(

(εN1)
2

N−2 , 1, N2

)
→ B

(
0, 1, N2

)
as ε→ 0, and

lim inf
α→0+

inf
dα∈Dα

[ dα

[log(e+ dα)]
α(N−2)

2

y′α(t)
]
≥ N1

N

2
B
(
0, 1,

N

2

)
= N1.

This completes the proof. �

Proof of Theorem 1.3. (i) First we prove (1.8). From (5.2), (2.3), (2.8) and (2.13)
we can write

lim inf
α→0+

inf
dα∈Dα

[ dα

[log(e+ dα)]α
N−2

2

uα(r)
]
≥ [N(N − 2)]

N−2
2

( 1

rN−2
− 1

RN−2

)
.

From (1.6) we deduce that

lim inf
α→0+

inf
dα∈Dα

[log(e+ dα)]
1
2 +αN+2

4

√
αdα

≥

√
1

L(N,R)
.

Multiplying both inequalities, we deduce that

lim inf
α→0+

inf
dα∈Dα

[
[log(e+ dα)]

1
2−α

N−6
4

uα(r)√
α

]
≥ L̃(N,R)

( 1

rN−2
− 1

RN−2

)
.

(ii) Next we prove (1.9). From (5.1), (2.3), (2.8) and (2.13), we can write

lim sup
α→0+

sup
dα∈Dα

[ dα

[log(e+ dα)]α
N
2

uα(r)
]
≤ [N(N − 2)]

N−2
2

( 1

rN−2
− 1

RN−2

)
.

From (1.7) we deduce

lim sup
α→0+

sup
dα∈Dα

[log(e+ dα)]α
N−4

4

dα

1√
α
≤

√
2∗

L(N,R)
.

Multiplying both inequalities we deduce

lim sup
α→0+

sup
dα∈Dα

[ 1

[log(e+ dα)]α
N+4

4

uα(r)√
α

]



EJDE-2020/114 ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS 17

≤
√

2? L̃(N,R)
( 1

rN−2
− 1

RN−2

)
.

This completes the proof. �

Acknowledgments. Rosa Pardo is supported bythe Spanish Ministerio de Ciencia
e Innovación (MICINN) under Projects MTM2016-75465 and PID2019-103860GB-
l00, and by the Grupo de Investigación CADEDIF 920894, UCM.

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Rosa Pardo
Universidad Complutense de Madrid, 28040 Madrid, Spain

Email address: rpardo@ucm.es

Arturo Sanjuán

Universidad Distrital Francisco José de Caldas, Bogotá, Colombia
Email address: aasanjuanc@udistrital.edu.co


	1. Introduction and main results
	2. Basic lemmas
	3. Further estimates and proof of Theorem 1.1
	4. Proof of Theorem 1.2
	5. Proof of Theorem 1.3
	Acknowledgments

	References