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Nonlinear Analysis 165 (2017) 102–120
Contents lists available at ScienceDirect
Nonlinear Analysis
www.elsevier.com/locate/na
On a class of nonhomogeneous equations of Hénon-type:
Symmetry breaking and non radial solutions ✩ , ✩✩
Ronaldo Brasileiro Assunçãoc , Olimpio Hiroshi Miyagakia ,
Gilberto de Assis Pereirac , Bruno Mendes Rodriguesb, *
a
Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, 36.036-000,
Brazil
b
Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, MG, 35.400-000, Brazil
c
Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 30.123-970,
Brazil
article
info
Article history:
Received 16 May 2017
Accepted 30 September 2017
Communicated by Enzo Mitidieri
MSC:
35B07
35J20
35J60
35J70
Keywords:
Hénon equation
Degenerate operator
Non radial solutions
abstract
In this work we study the following Hénon-type equation
(
)
⎧
p−2
⎪
⎨− div |∇u| ap ∇u = |x|β f (u),
in B;
⎪
⎩u > 0,
in B;
on ∂B;
|x|
u = 0,
{
}
where B := x ∈ RN ; |x| < 1 is a ball centered at the origin, the parameters
p+pβ
verify the inequalities 0 ≤ a < N p−p , N ≥ 4, β > 0, 2 ≤ p < NN
, and the
−p(a+1)
nonlinearity f is nonhomogeneous. By minimization on the Nehari manifold, we
prove that for large values of the parameter β there is a symmetry breaking and
non radial solutions appear.
© 2017 Elsevier Ltd. All rights reserved.
1. Introduction
In this work, we study a variant of the following problem
{
α
−∆u = |x| f (u), in B;
u = 0,
on ∂B;
(P)
where B is the open unit ball of RN centered at the origin with N ≥ 4.
✩
✩✩
*
The authors declare that they have no competing interests.
The authors declare that they contributed equally to the manuscript and that they read and approved the final draft of it.
Corresponding author.
E-mail address: [email protected] (B.M. Rodrigues).
https://doi.org/10.1016/j.na.2017.09.015
0362-546X/© 2017 Elsevier Ltd. All rights reserved.
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
103
Problem (P) was first studied in 1973 by Hénon [8], where he considered the case where f is a pure power
p−2
of the form f (t) = |t| t to model spherically symmetric galaxies; since then, this class of equations is
known as Hénon-type equations.
In 1982, Ni [14] proved that the problem −∆u = b(|x|)f (x), with Dirichlet boundary condition on the
unit ball, has at least one positive solution in the case where b verifies some suitable conditions at the origin
and at infinity, and f is not a pure power, i.e., it is nonhomogeneous, but is bounded by a power with
well-defined exponent of growth at infinity. Afterwards, Smets, Su and Willem [18] considered problem (P)
with a pure power nonlinearity f (t) = tp−1 , N ≥ 2, and 2 < p < 2∗ := 2N/(N − 2); they showed that
there exists α∗ > 0 such that α > α∗ a non radial ground state solution appears, which is a re-calling of the
function that achieves the infimum
∫
2
|∇u| dx
∫ Bα p
S = inf
,
2/p
u∈H 1 (B) ( B |x| |u| dx)
0
u̸=0
H01 (B)
where
is the Sobolev space. See Badiale and Serra [2], Serra [17], and Hirano [9] for more results
on multiplicity of solutions; see also Byeon and Wang [4] for a result on the asymptotic behavior of the
ground states; and see Kolonitskii [10] and Yang [12] for multiplicity results related to a quasilinear elliptic
problem involving the p-Laplacian operator associated to problem (P); moreover, for a Hénon-type system,
the reader is referred to the paper by Wang and Yang [20].
In 2009, Carrião, de Figueiredo and Miyagaki in [5] studied the following Hénon-type quasilinear elliptic
equation with a singularity on the differential operator,
(
)
⎧
p−2
|∇u| ∇u
⎪
β
q−2
⎪
⎪
= |x| |u| u, in B;
ap
⎨− div
|x|
⎪
⎪
in B;
⎪
⎩u > 0,
u = 0,
on ∂B;
where −∞ < a < (N − p)/p; N ≥ 3, β > 0 and 2 ≤ p < q < [(N + β)p]/[N − p(a + 1)].
They proved existence and multiplicity results, including the existence of a non radial solution, in the case
q ∈ (p, q ∗ ), where
⎧
(N + 2)p
⎪
⎪
⎨ N − 2p + 2 , if N is even;
q ∗ = q ∗ (N ) =
⎪
([N/2] + 2) p
⎪
⎩
, if N is odd.
[N/2] − p + 2
More recently, in 2014 Badiale and Cappa [1] studied problem (P) and proved the existence of a non
radial solution in the case where the nonlinearity f is not a pure power and with the range of growth
including supercritical exponents. This seems to be the first time this type of nonhomogeneous perturbation
appears in the literature associated to the existence of non radial solutions to Hénon-type equations. They
proved the existence of solutions as minima, on the Nehari manifold, of the energy functional associated in
a natural way to this class of problems. First, an estimate of “radial critical levels” was made; then another
estimate of “non radial critical levels” was achieved; finally, they showed that, for the parameter α large
enough, these critical levels are distinct. Motivated by the results in [1] with respect to the fact that the
majority of the papers dealing with non radial solutions of Hénon-type equations consider only pure powers
as nonlinearities, and also by [5] with respect to the presence of a singularity on the differential operator, in
this work we study the following class of problems
(
)
⎧
p−2
|∇u|
∇u
⎪
β
⎪− div
⎪
= |x| f (u), in B;
ap
⎨
|x|
(1)
⎪
⎪
u
>
0,
in
B;
⎪
⎩
u = 0,
on ∂B;
104
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
{
}
where B := x ∈ RN : |x| < 1 is the unit ball centered at the origin, 0 ≤ a < (N − p)/p, N ≥ 4, β > 0 and
2 ≤ p < [(N + β)p]/[N − p(a + 1)].
Note that, when we take a = 0 and p = 2, the problem (1) is reduced to the problem studied by Badiale
and Cappa in [1]. Therefore, in this work we are generalizing the Laplacian operator studied by Badiale and
Cappa. To prove our results, borrow some ideas and some results from [1,5,19].
To state our results, we first define l such that 2 ≤ p ≤ N − l ≤ l, we consider a such that
max {2p, p(a + 1)} ≤ N , and we define the critical exponent as
⎧
(N + 2)p
⎪
⎪
⎨ N − 2p + 2 , if N is even;
p∗ (N ) =
⎪
([N/2] + 2) p
⎪
⎩
, if N is odd.
[N/2] − p + 2
We show that problem (1) admits both radial and non radial solutions under the following hypotheses
on the nonlinearity f ;
(f1 ) f is a locally Hölder continuous function, f (z) ≥ 0 for all z > 0, f (z) = 0 for all z < 0, f (z) = o(z p−1 )
(z)
for z → 0; moreover limz→∞ zfp−1
= ∞;
}
{
2
p N
;
(f2 ) |f (z)| ≤ c(1 + |z|)q−1 for all z, where p < q < min p∗ (N ), (N −p)(p−1)
∫t
(f3 ) there exists τ > p such that tf (t) ≥ τ F (t) for all t ∈ R, where F (t) := 0 f (s)ds;
(f4 ) there exist µ1 , µ2 > p, such that, for all t ∈ [0, 1] and for v ≥ 0 we have f (tv) ≥ tµ1 −1 f (v) and for all
t ≥ 1 and for v ≥ 0 we have f (tv) ≥ tµ2 −1 g(v), where g(·) is a non-negative continuous function on R,
such that g(0) = 0; moreover, we have the following inequality
p2
µ1 − µ2
< N − l.
(µ1 − p)(µ2 − p)
(2)
We also denote by ma,β,r and by mla,β , respectively, the radial and the non radial critical levels of the
energy functional associated to problem (1).
Our main results read as follows.
Theorem 1.1. Under hypotheses (f 1)–(f 4), for
one non-negative radial solution to problem (1).
(N +β)p
N −p(a+1)
{
}
N p2
≥ max p∗ (N ), (N −p)(p−1)
there exists at least
Theorem 1.2. Under hypotheses (f 1)–(f 4), problem (1) has at least one non-negative, non radial solution
for the parameter β large enough. Moreover, mla,β < ma,β,r as β → +∞.
We would like to explain some words to the papers by Clément, de Figueiredo, and Mitidieri [6] and
Bozhkov and Mitidieri [3], which have been treated the problem (1), obtaining results related to us. In their
papers, they study Eq. (1) in the radial setting, with nonlinearity f as a polynomial. But in this setting,
they were able to involve a large class of the nonlinear operators, such as, p-Laplacian and k-Hessian.
Among their results , by using minimization arguments, in [6] is obtained the existence of radial solution
result, extending that result in [14] for more general nonlinear equation. While in [3], by applying the
fibering method, which was introduced by Pohazaev in [15] (see [7] for application), they get the existence,
multiplicity and non existence results of radial solutions, considering f as a polynomial which the coefficients
of this polynomial can change sign. In both arguments the homogeneity is essential to conclude the study.
The main improvement is the following. The problem (1) is nonhomogeneous, and in Theorem 1.1 we treat
the problem (1) with a general nonlinearity, while in Theorem 1.2 we obtained the existence of non radial
solution.
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
105
This work is organized as follows: in Section 2, we introduce an auxiliary problem and show that its radial
solutions are also solutions to problem (1); and in Section 3 and in Section 4, we estimate both the radial
critical level ma,β,r as well as the non radial critical level mla,β and we show that, for the parameter β large
enough, they are distinct; this guarantees our multiplicity result.
2. Proof of Theorem 1.1
A positive, radially symmetric function u(x) = u(|x|) is a solution to problem (1) if, and only if, the
function u(r) with r := |x| is a solution to problem
⎧ (
)′
p−2 ′
⎪
N −1−ap ′
⎪
−
r
|u
(r)|
u
(r)
= rβ+N −1 f (u(r)), in (0, 1);
⎪
⎨
u(r) > 0,
⎪
⎪
⎪
⎩
u(1) = u′ (0) = 0.
(3)
in (0, 1);
1,p
We denote by XN
−1−ap (0, 1) the Banach space of absolutely continuous radial functions u : [0, 1] −→ R,
′
such that u(1) = u (0) = 0 endowed with the norm ∥·∥X 1,p
defined by
N −1−ap
1
(∫
∥u(r)∥X 1,p
r
:=
N −1−ap
N −1−ap
) p1
p
′
< ∞.
|u (r)| dr
0
Let Lqβ+N −1 (0, 1) be the Banach space of Lebesgue measurable functions u : [0, 1] −→ R with the norm
defined by
|·|Lq
β+N −1
(∫
|u(r)|Lq
β+N −1
1
r
:=
β+N −1
) 1q
q
|u(r)| dr
< ∞.
0
1,p
Denote by I : XN
−1−ap (0, 1) −→ R the energy functional associated to problem (3), and defined by
I(u) := ωN
( ∫ 1
)
∫ 1
1
p
rN −1−ap |u′ (r)| dr −
rβ+N −1 F (u(r))dr .
p 0
0
Using standard results, we have
(∫ 1
∫
p−2
′
I (u)(v) = ωN
rN −1−ap |u′ (r)| u′ (r)v ′ (r)dr −
0
1
r
β+N −1
)
f (u(r))v(r)dr
0
1,p
for every function v ∈ XN
−1−ap (0, 1), where ωN is the measure of the surface of the unit ball. Therefore,
critical points for the energy functional I are weak solutions to the auxiliary problem (3) and, by our setting,
they are also solutions to problem (1).
In [5, Proposition 1.1], we have an important compactness result which guarantees that the space
1,p
p+pβ
XN −1−ap (0, 1) is compactly embedded in Lqβ+N −1 (0, 1) if q < NN
−p(a+1) . The next two lemmas assure
us that the functional I satisfies the geometry and also the compactness condition of the mountain-pass
theorem by Ambrosetti and Rabinowitz [16].
1,p
Lemma 2.1. The functional I satisfies the Palais–Smale condition in XN
−1−ap (0, 1) on the level c ∈ R.
1,p
Proof . Let {un } ⊂ XN
−1−ap (0, 1) be a Palais–Smale sequence for the functional I, i.e., {un } satisfies
(i) I(un ) → c;
106
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
(
)∗
1,p
(ii) I ′ (un ) → 0 in XN
(0,
1)
,
−1−ap
(
)∗
1,p
1,p
where XN
is the dual space of XN
−1−ap (0, 1)
−1−ap (0, 1).
1,p
We divide the proof of this lemma in two steps. First, we show that the sequence {un } ⊂ XN
−1−ap (0, 1)
1,p
is bounded; then, we prove that it is a Cauchy sequence in the Banach space XN
(0,
1).
−1−ap
Step 1. Since {un } is a sequence (P S)c , by hypothesis (f3 ) we have
1
1
c + 1 ≥ I(un ) = I(un ) − I ′ (un )(un ) + I ′ (un )(un )
τ
τ
)
(
∫ 1
1 1
p
∥un ∥XN −1−ap − ωN
rβ+N −1 F (un )dr
−
= ωN
p τ
0
∫
1
ωN 1 β+N −1
r
f (un )un dr + I ′ (un )(un )
+
τ 0
τ
)
(
1
1 1
p
−
∥un ∥XN −1−ap + I ′ (un )(un ),
≥ ωN
p τ
τ
for n ∈ N large enough. Since τ > p, the previous inequality implies that the sequence {un } is bounded in
1,p
XN
−1−ap (0, 1).
Step 2. It follows from hypotheses (f1 ) and (f2 ) that, for all ε > 0, there exists a cε > 0, such that
p−1
f (u(r)) ≤ ε|u(r)|
+ cε |u(r)|
q−1
.
Moreover, using in the inequality
)
⎧(
p−2
p−2
⎪
⎨ |ξ| ξ − |η| η (ξ − η) ,
p
|ξ − η| ≤ [(
)
]p
2−p
⎪
⎩ |ξ|p−2 ξ − |η|p−2 η (ξ − η) 2 [|ξ|p + |η|p ] 2 ,
(4)
if p ≥ 2
(5)
if 1 < p < 2,
for all ξ, η ∈ RN (see [11]), for the case p ≥ 2 we obtain
∫ 1
⏐
⏐p
p
= ωN
ωN ∥ui − uj ∥ 1,p
rN −1−ap ⏐u′i − u′j ⏐ dr
XN −1−ap
∫0 1
(
⏐ ⏐p−2 ) (
)
p−2
≤ ωN
rN −1−ap |u′i | u′i − ⏐u′j ⏐ u′j u′i − u′j dr
0
′
≤ |I (ui )(ui − uj )| + |I ′ (uj )(ui − uj )|
⏐
⏐∫ 1
⏐
⏐
β+N −1
⏐
+ωN ⏐
r
(f (ui ) − f (uj )) (ui − uj )dr⏐⏐
(6)
0
:= I1 + I2 + I3 .
(
)
(
)
Note that I1 = o ∥un ∥XN −1−ap and I2 = o ∥un ∥XN −1−ap follow immediately from the fact that the
1,p
sequence {un } is (P S)c . It follows from inequality
( (4) and from
) the compact embedding XN −1−ap (0, 1) ↪→
p+pβ
Lqβ+N −1 (0, 1) with q < NN
−p(a+1) that I3 = o ∥un ∥XN −1−ap . Therefore, {un } is a Cauchy sequence and
the functional I satisfies the Palais–Smale condition. □
Lemma 2.2. Let I be the functional associated to problem (3); then
⏐
⏐
(i) there are positive constants ρ, α such that I ⏐
≥ α,
∂Bρ
1,p
(ii) there is e ∈ XN
−1−ap (0, 1) \ Bρ such that I(e) < 0.
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
107
Proof . (i) Let ε > 0. Then, by the inequality (4), by the hypothesis (f3 ) and by the compact embedding
1,p
q
N p+pβ
XN
−1−ap (0, 1) ↪→ Lβ+N −1 (0, 1) with q < N −p(a+1) , we have
1
1
I(u) =
ωN
p
∫
1
≥
p
∫
≥
≥
1
p
(
1
p
rN −1−ap |u′ | dr −
∫
0
1
rβ+N −1 F (u)dr
0
1
r
N −1−ap
0
1
|u | dr −
τ
′ p
1
∫
rβ+N −1 f (u)udr
0
∫
∫
ε 1 β+N −1 p
cε 1 β+N −1 q
p
rN −1−ap |u′ | dr −
r
|u| dr −
r
|u| dr
τ 0
τ 0
0
)
1
c1
cε c2
p
q
−ε
∥u∥XN −1−ap −
∥u∥XN −1−ap .
p
τ
τ
∫
1
Therefore, fixing ε > 0 small enough in the previous inequality, we can find ρ > 0 and α > 0 such that, for
1,p
every u ∈ XN
−1−ap (0, 1) with ∥u∥XN −1−ap = ρ, the inequality I(u) ≥ α is valid, since p < q.
(ii) By hypotheses (f2 ) and (f3 ) it is possible to find constants c1 , c2 > 0, such that
q
p
c1 |u| − c2 |u| ≤ F (u).
1,p
Let u ∈ XN
−1−ap (0, 1) be such that ∥u∥N −1−ap = 1; then, for any t > 1, we have
1
1
I(tu) = tp −
ωN
p
1
∫
rβ+N −1 F (tu)dr
0
1
≤ tp + c2 tp
p
∫
1
r
β+N −1
p
q
∫
|u| dr − c1 t
0
1
q
rβ+N −1 |u| dr.
0
Since p < q, by previous inequality we can conclude that I(tu) → −∞ as t → +∞. So, there is
1,p
e ∈ XN
−1−ap (0, 1) \ Bρ , such that I(e) < 0. □
Proof of Theorem 1.1. By Lemmas 2.1 and 2.2, all the assumptions of the Mountain Pass Theorem
1,p
in [16] are satisfied. Hence, we deduce the existence of a radial function u∗ ∈ XN
−1−ap (0, 1), which is a weak
∗
solution to problem (3); moreover I(u ) ≥ α > 0. □
3. “Radial critical levels” and their estimates
1,p
To begin the proof of Theorem 1.2, we denote by Ia,β : XN
−1−ap (0, 1) → R the functional associated to
problem (1), defined by
∫
∫
1
−ap
p
β
Ia,β (u) :=
|x|
|∇u| dx −
|x| F (u)dx
p B
B
and we introduce the following Nehari manifolds
{
}
∫
∫
−ap
p
β
1,p
Na,β,r := u ∈ XN
(0,
1)
\
{0}
:
|x|
|∇u|
dx
=
|x|
f
(u)udx
.
−1−ap
B
B
If we denote by
ma,β,r :=
inf
u∈Na,β,r
Ia,β (u)
the “radial critical level′′ of the functional Ia,β , then by Theorem 1.1 ma,β,r is attained. The next results
will help us obtain an estimate from below for ma,β,r .
108
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
1,p
Define now α ≡ p1 (N − p) and let u ∈ XN
−1−ap (B) be such that
∫
|x|
−α
p
|∇u| dx < +∞.
B
−b
Note that if b < α, then |x|
−α
< |x| , because |x| < 1 in B. Hence,
∫
∫
−b
p
−α
p
|x| |∇u| dx ≤
|x| |∇u| dx < +∞.
B
B
1,p
N
If we extend the function u setting u = 0 on RN \ B, we have u ∈ XN
−1−ap (R ) and
∫
|x|
−b
∫
p
|∇u| dx ≤
|x|
RN
−α
p
|∇u| dx < +∞.
RN
Now, we need the following result proved by Su and Wang [19, Lemma 2.1].
Lemma 3.1 (Weighted
Version
of the Radial Lemma). Let ξ ∈ R be such that 1 < p < N + ξ. Let
(
)
ξ
1,p
N
∞
X = XN
R
,
|x|
dx
the
completion
of C0,r
(R) under the norm
−1−ap
) p1
|x| |∇u| dx .
(∫
∥u∥r,ξ :=
ξ
p
RN
Then there is a constant c > 0, such that for all u ∈ X it holds
−
|u(x)| ≤ c|x|
N +ξ−p
p
∥u∥r,ξ .
1,p
For α and b previously defined, we apply Lemma 3.1 to ξ = −b and u ∈ XN
−1−ap (B), setting u = 0 in
RN \ B. Hence, there is a constant c = cb > 0 such that, for a.e. x ∈ RN , it holds
−
|u(x)| ≤ c|x|
N −b−p
p
(∫
|x|
−b
) p1
|∇u| dx
p
RN
−
= c|x|
N −b−p
p
(∫
−b
|x|
) p1
|∇u| dx
p
B
−
≤ c|x|
N −b−p
p
(∫
−α
|x|
p
) p1
|∇u| dx
.
B
Now, we prove the following lemma.
∫
−α
p
1,p
Lemma 3.2. Let u ∈ XN
|∇u| dx < +∞ and p < q <
−1−ap (B) be such that B |x|
pN
b = N − p − q then there exists a constant c = cb > 0, such that
(∫
) pq
∫
−b
p
|u| dx
≤ cb
|x| |∇u| dx.
q
B
B
In particular, since b < α we have
(∫
B
) 1q
(∫
) p1
−α
p
|u| dx
≤c
|x| |∇u| dx .
q
B
p2 N
(N −p)(p−1) .
If
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
109
2
p N
; moreover, the value of b
Proof . First we note that b = N − p − Nqp < α = N p−p because q < (N −p)(p−1)
Np
implies that q = N −b−p .
Since u(1) = 0, using the integration by parts together with Hölder’s inequality, we have
∫
∫ 1
q
q
|u| dx = ωN
rN −1 |u(r)| dr
B
0
⏐r=1 qω ∫ 1
1 N
N
q⏐
q−2
−
rN |u(r)| u(r)u′ (r)dr
r |u(r)| ⏐
N
N 0
r=0
∫
qωN 1 N
q−1
r |u(r)|
|u′ (r)| dr
N 0
∫
N −b−1
qωN 1 N − N −b−1
q−1
p
r
|u(r)|
|u′ (r)| r p dr
N 0
[∫ 1 (
] p−1
) ] p1 [∫ 1 (
) p
p
N −b−1 p
qωN
q−1 p−1
N − N −b−1
′
p
p
|u (r)| r
dr
r
|u(r)|
dr
N
0
0
] p−1
[∫ 1
] p1 [∫ 1
p
q−1
b+1
pωN
p N −1
p p−1
N + p−1
−b ′
dr
|u(r)|
.
r
r |u (r)| r
dr
N −b−p 0
0
= ωN
≤
=
≤
=
Using Lemma 3.1 we have
∫
∫ 1
q−1
b+1
p
rN + p−1 |u(r)| p−1 dr =
1
q
q−p
b+p
rN −1 |u(r)| r p−1 |u(r)| p−1 dr
0
0
q−p
p−1
1
∫
≤ c1
N −b−p q−p
q b+p
rN −1 |u(r)| r p−1 r− p p−1
0
p
rN −1−b |u′ (r)| dr
0
(∫
−b
|x|
= c2
1
(∫
q−p ∫
) p1 p−1
|∇u| dx
1
p
B
q
rN −1 |u(r)| dr,
0
b+p
q−p
because p−1
− N −b−p
p
p−1 = 0.
Therefore, by the two previous inequalities, we deduce that
∫
(∫
q
−b
|u| dx ≤ c3
B
|x|
) p1
p
|∇u| dx
B
⎡
⎤ p−1
p
q−p ∫
(∫
) p1 p−1
−b
p
q
⎣
|x| |∇u| dx
|u| dx⎦
B
B
(∫
−b
|x|
= c3
(∫
) p−1
) p1 (∫
) q−p
p
p2
q
−b
p
|x| |∇u| dx
|u| dx
|∇u| dx
p
B
B
B
) q2 (∫
) p−1
p
p
−b
p
q
|x| |∇u| dx
|u| dx
;
(∫
= c3
B
B
therefore,
(∫
) p1
(∫
)1− p−1
p
q
|u| dx
=
|u| dx
q
B
B
(∫
≤ c3
−b
|x|
B
) q2
p
|∇u| dx
.
p
q−p
) p1 p−1
dr
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
110
It follows immediately from the previous inequality that
(∫
) pq
∫
q
−b
p
≤ c4
|u| dx
|x| |∇u| dx.
B
□
B
We now introduce the objects we need to work in a Nehari frame. For this, assuming β > N and taking
α = N p−p as previously defined, we define the space
{
}
∫
−α
p
1,p
H := v ∈ XN −1−ap (0, 1) :
|x| |∇v| dx < +∞ ,
B
the functional J : H → R by
J(v) =
1
p
∫
−α
|x|
∫
p
|∇v| dx −
B
F (v)dx,
B
the Nehari manifold
{
M :=
∫
1,p
v ∈ XN
−1−ap (0, 1) \ {0} :
|x|
−α
}
f (v)vdx
∫
p
|∇v| dx =
B
B
and
m′ := inf {J(v) : v ∈ M } .
Our first result regarding the Nehari manifold is the following.
Lemma 3.3. M ̸= ∅.
Proof . Let v ∈ C0∞ (B \ {0}) be a function different from zero such that
ψ : (0, +∞) → R be a continuous function defined by
∫
∫
−α
p
ψ(t) := tp
|x| |∇v| dx −
f (tv)tvdx.
B
−α
∫
B
|x|
p
|∇v| dx < +∞, and let
B
We will study their behavior as t → 0+ and as t → +∞. By hypotheses (f1 ) and (f2 ), for all ε > 0 we can
find a constant cε > 0 such that
⏐∫
⏐
∫
∫
⏐
⏐
p
q
q
⏐ f (tv)tvdx⏐ ≤ εtp
|v|
dx
+
c
t
|v| dx.
(7)
ε
⏐
⏐
B
B
B
Since q > p, by the inequality (7) we can conclude that
∫
f (tv)tvdx = o(tp ), as t → 0+ .
B
In this way, we have
p
∫
−α
|x|
ψ(t) = t
p
|∇v| dx + o(tp ) as t → 0+ .
B
Hence ψ(t) > 0 as t → 0+ . On the other hand, by hypothesis (f3 ) there exists a constant c > 0 such that
f (t)t ≥ ctτ for all t ≥ 1. So,
∫
∫
−α
p
τ
p
τ
ψ(t) ≤ t
|x| |∇v| dx − ct
|v| dx.
(8)
B
B
Therefore, by the inequality (8), we conclude that ψ(t) → −∞ as t → +∞, because τ > p. Since ψ is a
continuous function, then there exists t0 ∈ (0, +∞), such that ψ(t0 ) = 0, i.e., t0 v ∈ M . □
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
111
In the next result we will prove that m′ is positive.
Lemma 3.4. m′ > 0.
Proof . We prove first that m′ ≥ 0. Indeed, from hypothesis (f3 ) we deduce that, if v ∈ M , it holds
∫
∫
1
−α
p
|x| |∇v| dx −
J(v) =
F (v)dx
p B
B
∫
∫
1
1
−α
p
|x| |∇v| dx −
f (v)vdx
≥
p B
τ B
(
)∫
1 1
−α
p
=
−
|x| |∇v| dx ≥ 0.
p τ
B
Now, we will prove that m′ > 0. Taking again v ∈ M and noting that 1 < p < q <
Lemma 3.2, we have the following inequality
p2 N
(N −p)(p−1) ,
then by
(∫
) p1
) 1q
−α
p
q
≤c
|x| |∇v| dx .
|v| dx
(∫
B
B
Contrary to the Laplace operator, the p-Laplacian spectrum has not been proved to be discrete. In [13], the
first and the second eigenvalues of the p-Laplacian operator are described. Let λ1 be the first eigenvalue of
the operator −∆p . Using hypotheses (f1 ) and (f2 ), we can choose a constant c1 > 0 such that
|f (t)t| ≤
1
p
q
λ1 |t| + c1 |t| , ∀t ∈ R.
p
Hence, we get
∫
−α
|x|
∫
p
∫
|∇v| dx =
f (v)vdx ≤
|f (v)v| dx
B
∫
∫
λ1
p
q
≤
|v| dx + c1
|v| dx
p B
B
(∫
) pq
∫
1
p
−α
p
q
|∇v| dx + c1 c
|x| |∇v| dx
≤
p B
B
(∫
) pq
∫
1
−α
p
−α
p
q
≤
|x| |∇v| dx + c1 c
|x| |∇v| dx ,
p B
B
B
B
then
∫
|x|
−α
[(
p
|∇v| dx ≥
B
for all v ∈ M. Since J(v) ≥
(
1
p
−
1
τ
)∫
B
|x|
−α
1
1−
p
)
1
c1 cq
p
] q−p
>0
p
|∇v| dx for all v ∈ M , then m′ > 0.
□
1,p
Let uβ ∈ XN
−1−ap (0, 1) be the solution of (1), such that uβ ∈ Na,β,r and
Ia,β (uβ ) = ma,β,r =
inf
u∈Na,β,r
Ia,β (u).
As in [18], we define vβ (x) = vβ (ρ) = uβ (ρs ), where ρ = |x| and s =
N
β+N ,
so that s → 0 as β → ∞.
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
112
Therefore, by making some changes of variables, we have
∫
∫
β
1
rβ+N −1 f (uβ (r))uβ (r)dr
|x| f (uβ )uβ dx = ωN
B
0
∫
1
ρs−1 ρs(β+N −1) f (vβ (ρ))vβ (ρ)dρ
= sωN
(9)
0
∫
1
N −1
= sωN
ρ
f (vβ (ρ))vβ (ρ)dρ
∫ 0
= s
f (vβ (x))vβ (x)dx
B
and
∫
|x|
−ap
∫
1
⏐
⏐p
r−ap rN −1 ⏐u′β (r)⏐ dr
0
∫ 1
⏐p
⏐
1
ρ−aps ρs(N −1) ρ−(s−1)(p−1) ⏐vβ′ (ρ)⏐ dρ
= p−1 ωN
s
∫0 1
⏐
⏐p
1
= p−1 ωN
ρ−γ ρN −1 ⏐vβ′ (ρ)⏐ dρ
s
∫ 0
1
−γ
p
|x| |∇vβ (x)| dx,
= p−1
s
B
p
|∇uβ | dx = ωN
B
(10)
where γ := aps + (N − p)(1 − s) > 0. Note that by repeating the calculations performed in (9) for F (uβ ),
we can also obtain the following relation
∫
∫
β
|x| F (uβ )dx = s
F (vβ (x))dx.
(11)
B
B
N −p
N
Notice that, for fixed N and β > p−1
, we have s < p−1
= α. Then, for |x| < 1, we have
p , so that γ >
p
−γ
−α
|x| > |x|
and, consequently,
∫
∫
−α
p
−γ
p
|x| |∇vβ | dx ≤
|x| |∇vβ | dx < +∞.
B
B
Now we define the space
{
v∈
Hs :=
1,p
XN
−1−ap (0, 1)
∫
|x|
:
−γ
}
p
|∇v| dx < +∞ ,
B
the functional Js : Hs → R by
Js (v) =
1
p
∫
|x|
−γ
∫
p
|∇v| dx −
B
F (v)dx,
B
the Nehari manifold
{
Ms :=
1,p
v ∈ XN
−1−ap (0, 1) \ {0} :
∫
|x|
−γ
p
∫
|∇v| dx =
B
}
f (v)vdx
B
and
ms := inf {Js (v) : v ∈ Ms } .
Lemma 3.5. If β > N and s =
N
β+N ,
then there exists τ > p, such that ms ≥
τ −p ′
τ m .
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
Proof . If v ∈ Ms , then
−γ
∫
B
|x|
∫
113
p
|∇v| < +∞ and
∫
∫
−α
p
−γ
p
|x| |∇v| dx ≤
|x| |∇v| dx =
f (v)vdx.
B
B
(12)
B
Since
p
∫
|x|
ψ(t) := t
−α
∫
p
|∇v| dx −
B
f (tv)tvdx,
B
then by the relation (12), we have ψ(1) ≤ 0 and, by the same arguments used in the proof of Lemma 3.3,
we get that ψ(t) > 0 as t → 0+ . So, by the continuity of the function ψ, there exists ts ∈ (0, 1], such that
ψ(ts v) = 0, i.e., ts v ∈ M . Therefore, by hypothesis (f3 ) there exists τ > p such that
∫
∫
1
−α
p
|x| |∇(ts v)| dx −
F (ts v)dx
m′ ≤ J(ts v) =
p B
B
∫
∫
tp
1
−α
p
−γ
p
|x| |∇v| dx ≤
|x| |∇v| dx
≤ s
p B
p B
(
)∫
τ
1 1
−γ
p
=
−
|x| |∇v| dx
τ −p p τ
B
[(
)∫
) ]
∫ (
τ
1 1
1
−γ
p
≤
−
|x| |∇v| dx +
f (v)v − F (v) dx
τ −p
p τ
τ
B
B
[ ∫
]
∫
τ
1
−γ
p
=
|x| |∇v| dx −
F (v)dx
τ −p p B
B
τ
Js (v),
=
τ −p
i.e.,
τ −p ′
τ m
≤ Js (v) for all v ∈ Ms . Therefore, we have ms ≥
τ −p ′
τ m .
□
Proposition 3.6. There exists c > 0 such that
ma,β,r ≥ cβ
µ1 (p−1)+p
µ1 −p
, as β → +∞.
Proof . Let us first see that there exists tβ > 0 such that tβ vβ ∈ Mβ , i.e.,
∫
∫
−γ
p
tpβ
|x| |∇vβ | dx =
f (tβ vβ )tβ vβ dx.
B
B
In fact, let the function φ : (0, +∞) → R be defined by
∫
∫
−γ
p
φ(t) := tp
|x| |∇vβ | dx −
f (tvβ )tvβ dx.
B
B
Then, by the relations (9) and (10), we have
∫
∫
−γ
p
φ(1) =
|x| |∇vβ | dx −
f (vβ )vβ dx
B
B
∫
∫
1
β
−ap
p
|x| f (uβ )uβ dx
= sp−1
|x|
|∇uβ | −
s B
B
[ ∫
]
∫
1 p
−ap
p
β
=
s
|x|
|∇uβ | −
|x| f (uβ )uβ dx
s
B
B
[ ∫
]
∫
1 p
−ap
p
−ap
p
=
s
|x|
|∇uβ | −
|x|
|∇uβ | dx
s
B
B
∫
1 p
−ap
p
= (s − 1)
|x|
|∇uβ | dx < 0.
s
B
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
114
On the other hand, using the same arguments in Lemma 3.3, we can show that
∫
−γ
p
φ(t) = tp
|x| |∇vβ | dx + o(tp ) as t → 0+ ;
B
+
hence, φ(t) ≥ 0 as t → 0 . So, by the continuity of the function φ there exists tβ ∈ (0, 1), such that
φ(tβ ) = 0, i.e., tβ vβ ∈ Ms .
From hypothesis (f4 ), we get
∫
∫
∫
−γ
p
tpβ
|x| |∇vβ | dx =
f (tβ vβ )tβ vβ dx ≥ tµβ
f (vβ )vβ dx.
(13)
B
B
B
Again using relations (9) and (10), together with relation (13), we have
∫
∫
−ap
p
−γ
p
sp−1 B |x|
|∇uβ | dx
|x| |∇vβ | dx
µ1 −p
B∫
= 1∫
= sp ,
tβ
≤
β
f (vβ )vβ dx
|x| f (uβ )vβ dx
B
s
B
because uβ ∈ Ma,β,r . So we can write
p
tβ ≤ s µ1 −p .
(14)
Therefore, by Lemma 3.4, by hypothesis (f3 ) and by the relations (9), (10), (11), and (14), we have
∫
∫
tpβ
τ −p ′
−γ
p
m ≤ ms ≤ Js (tβ vβ ) =
|x| |∇vβ | dx −
F (tβ vβ )dx
τ
p B
B
∫
tpβ
−γ
p
|x| |∇vβ | dx
≤
p B
(
) ∫
τ
1 1 p
−γ
p
=
−
t
|x| |∇vβ | dx
τ −p p τ β B
[(
) ∫
) ]
∫ (
τ
1 1 p
1
−γ
p
≤
−
tβ
|x| |∇vβ | dx + tµβ 1
f (vβ )vβ − F (vβ ) dx
τ −p
p τ
τ
B
B
[(
) ]
)∫
(
∫
µ
(p−1)+p
1
1 1
τ
1
−ap
p
β
≤
s µ1 −p
−
f (uβ )uβ − F (uβ ) dx
|x|
|∇uβ | dx +
|x|
τ −p
p τ
τ
B
B
[ ∫
]
∫
µ
(p−1)+p
1
1
τ
−ap
p
β
=
s µ1 −p
|x|
|∇uβ | dx −
|x| F (uβ )dx
τ −p
p B
B
=
µ1 (p−1)+p
τ
s µ1 −p ma,β,r .
τ −p
Hence,
ma,β,r
)2 ( ) µ1 (p−1)+p
µ1 −p
1
≥ m
s
(
)2 (
) µ1 (p−1)+p
µ1 −p
N +β
′ τ −p
= m
τ
N
′
≥ cβ
(
τ −p
τ
µ1 (p−1)+p
µ1 −p
. □
4. “Non radial critical levels” and their estimates
The existence of a non radial solution to problem (1) is obtained by following some ideas in Carrião, de
Figueiredo and Miyagaki [5]. First, let us define the Banach space Da1,p (B) as the completion of the space
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
115
∫
p
−ap
p
C0∞ (B), with respect to the norm given by ∥u∥ := B |x|
|∇u| dx. Then, we introduce a subspace of
Da1,p (B), defined by
{
}
1,p
Da,l
(B) := u ∈ Da1,p (B) : u(y, z) = u(|y| , |z|), x = (y, z) ∈ Rl × RN −l ,
where 2 ≤ p ≤ N − l ≤ l and N ≥ max {2p, p(a + 1)} . In [5, Theorem 2.1] it is proved an important
1,p
compactness result which states that the embedding Da,l
(B) ↪→ Lqβ (B) is continuous if q ≤ pl and
N p+(l−1)(l+1−p)
,
β ≥ β0 := plpql , and it is compact for all q < pl and β ≥ β0 ; where pl := (l+1)p
l+1−p , ql :=
l+1
{ ∫
}
β
p
p
and Lβ (B) := u : B |x| |u| dx < ∞ . Arguing in a similar ways as we have already done in the proof
of Theorem 1.1, using the compactness result obtained in [5] together with the mountain-pass theorem of
Ambrosetti and Rabinowitz [16], we can prove the first part of Theorem 1.2, i.e., ensure the existence of at
least one non radial solution to problem (1).
Now, we will prove the second part of Theorem 1.2, i.e., we will show that the radial and non-radial
critical levels are distinct. To do this, we consider p2 ≤ l ≤ N − p and x = (y, z) ∈ Rl × RN −l , and we
introduce the following Nehari manifold
{
}
1,p
l
′
Na,β
:= u ∈ Da,l
(B) \ {0} : Ia,β
(u)u = 0 .
If we denote by
mla,β :=
Ia,β (u)
inf
l
u∈Na,β
the “non radial critical level” of the functional Ia,β , we will obtain an estimate from over for mla,β .
First, we will prove that mla,β is positive.
Proposition 4.1. mla,β > 0
l
Proof . For v ∈ Na,β
, we have
∫
−ap
∫
p
|x|
β
|∇v| dx =
|x| f (v)vdx.
B
B
By hypotheses (f1 ) and (f2 ), for every ε > 0 there exists a constant cε > 0 such that
|f (z)z| ≤ εz p + cε z q , ∀z ∈ R.
In [5, Theorem 2.2], it is proved the inequality
(∫
) pq
∫
−ap
p
|x| |v| dx
≤c
|x|
|∇v| dx
β
q
B
B
for some constant c > 0. Therefore, using the previous estimates, we obtain
∫
∫
∫
−ap
p
β
p
β
q
|x|
|∇v| dx ≤ ε
|x| |v| dx + cε
|x| |v| dx
B
B
B
∫
≤ c1 ε
−ap
|x|
p
(∫
|∇v| dx + c2 cε
B
|x|
−ap
) pq
p
|∇v| dx ,
B
p
where c1 and c2 are positive constants that do not depend on ε > 0. Since ∥v∥ =
we can write
p
p
q
∥v∥ ≤ c1 ε ∥v∥ + c2 cε ∥v∥ .
∫
B
|x|
−ap
p
|∇v| dx, then
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
116
So by choosing ε > 0, such that 1 − c1 ε > 0, we have
(
∥v∥ ≥
1 − c1 ε
c2 cε
1
) q−p
> 0.
l
By hypothesis (f3 ), there exists τ > p such that for all v ∈ Na,β
, we have
∫
∫
1
−ap
p
β
Ia,β (v) =
|x|
|∇v| dx −
|x| F (v)dx
p B
B
∫
∫
1
1
−ap
p
β
|x|
|∇v| dx −
|x| f (v)vdx
≥
p B
τ B
(
)
1 1
p
=
−
∥v∥
p τ
)(
) p
(
1 − c1 ε q−p
1 1
−
> 0.
≥
p τ
c2 cε
Therefore, we can conclude that mla,β > 0. □
We now prove the following proposition.
Proposition 4.2. There exists c > 0, such that
µ2 (p−1)+p
−N +l
µ2 −p
mla,β ≤ cβ
, as β → +∞.
Proof . Consider the following set
{
}
D := (λ, t) ∈ R2 : λ, t ≥ 0 and 0 ≤ λ2 + t2 < 1 .
1,p
For all u ∈ Da,l
(B), we define u(x) = u(|y| , |z|) = u(λ, t), where x = (y, z) ∈ Rl × RN −l with λ = |y| and
t = |z|. Therefore, by making this change of variables, we have
∫
∫
( 2
) −ap
−ap
p
p
|x|
|∇u| dx = c
λ + t2 2 |∇u(λ, t)| λl−1 tN −l−1 dλdt
(15)
B
D
and
∫
∫
β
|x| F (u)dx = c
B
(
λ2 + t2
) β2
F (u(λ, t))λl−1 tN −l−1 dλdt.
(16)
D
1,p
Consider the functional Ia,β on Da,l
. Using the polar coordinates in R2 : λ = ρ cos θ and t = ρ sin θ, then the
set D is equal to the set
{
π}
A = (ρ, θ) : 0 ≤ ρ < 1, 0 ≤ θ ≤
.
2
Defining v(ρ, θ) := u(ρ cos θ, ρ sin θ), we have
uλ = vρ cos θ −
2
so, |∇u| = u2λ + u2t = vρ2 +
∫
D
(
2
2
λ +t
) −ap
2
1 2
v .
ρ2 θ
1
1
sin θvθ and ut = vρ sin θ + cos θvθ ;
ρ
ρ
Hence,
p l−1 N −l−1
|∇u(λ, t)| λ
t
)p
∫ (
1 2 2 N −1−ap
2
dλdt =
vρ + 2 vθ
ρ
H(θ)dρdθ
ρ
B
(17)
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
117
and
∫
(
λ2 + t2
) β2
F (u(λ, t))λl−1 tN −l−1 dλdt =
∫
D
F (v(ρ, θ))ρβ+N −1 H(θ)dρdθ,
(18)
A
N −l−1
l−1
where H(θ) := (cos(θ)) (sin(θ))
.
Now we introduce a rectangle à ⊂ A, defined by
)
(
1 3
,
× (θ1 , θ2 ),
à :=
4 4
with 0 < θ1 < θ2 <
the set
π
2.
Consider a non-negative function ψ ∈ Co∞ (Ã) \ {0}; then for every ε > 0, we define
{
Ãε :=
(ρ, θ) ∈ R2 :
( )ε
( )ε
}
1
3
≤ρ≤
, εθ1 ≤ θ ≤ εθ2
4
4
and the function v ε ∈ C0∞ (Ãε ), given by
(
)
1 θ
v ε (ρ, θ) := ψ ρ ε ,
.
ε
Now we calculate Ia,β on a function uε , defined by
uε (x) = uε (|y| , |z|) = uε (ρ cos θ, ρ sin θ) = v ε (ρ, θ),
1,p
so that, uε ∈ C0∞ (B) ∪ Da,l
(B).
Set
ε :=
N
,
β+N
so that, ε → 0 as β → +∞.
1
Making the changes of variables r = ρ ε and φ = θε and noting that
)
)
(
(
1 θ
1 θ
1 1
1 1
vρε (ρ, θ) = ρ ε ψ1 ρ ε
and vθε (ρ, θ) = ρ ε ψ2 ρ ε
,
ε
ε
ε
ε
where ψi :=
∂ψ
∂si ,
and using the notation ψ = ψ(s1 , s2 ), we have that
∫
∫
β
|x| F (uε )dx = cε2
F (ψ)rN −1 H(εφ)drdφ,
B
∫
(19)
Ã
β
∫
f (ψ)ψrN −1 H(εφ)drdφ,
(20)
) p2
∫ (
1
ψ12 + 2 ψ22 r−ap r(ε−1)(N −p(a+1)) H(εφ)rN −1 drdφ,
r
Ã
(21)
|x| f (uε )uε dx = cε2
B
Ã
and
∫
−ap
|x|
B
p
|∇uε | dx = ε2−p
with (ε − 1)(N − p(a + 1)) < 0.
l
We claim that if ε → 0, then exists tε > 1 such that tε uε ∈ Na,β
.
Indeed, let h : (0, +∞) → R be a continuous function defined by
∫
∫
−ap
p
β
h(t) := tp
|x|
|∇uε | dx −
|x| f (tuε )tuε dx.
B
B
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
118
Rewrite the function h in the form
h(t) = tp
[∫
−ap
|x|
∫
ε p
|∇u | dx −
B
β
|x|
B
f (tuε )
(tuε )
p−1 (u
]
ε p
) dx
and applying hypothesis (f1 ), we can conclude that h(t) → −∞, as t → +∞.
Note that for ε small enough, there are positive constants c1 and c2 such that
c1 εN −l−1 ≤ H(εφ) ≤ c2 εN −l−1 .
(22)
Hence, using the relations (20)–(22), we have
∫
−ap
|x|
h(1) =
∫
ε p
β
|x| f (uε )uε dx
|∇u | dx −
B
B
)p
∫ (
1 2 2 −ap (ε−1)(N −p(a+1))
2−p
2
r
H(εφ)rN −1 drdφ
= ε
ψ1 + 2 ψ2 r
r
Ã
∫
2
− cε
f (ψ)ψrN −1 H(εφ)drdφ
Ã
[
≥ c3 εN −l−1
]
) p2
∫
∫ (
1
f (ψ)ψrN −1 drdφ > 0,
ε2−p
ψ12 + 2 ψ22 r−ap rN −1 drdφ − cε2
r
Ã
Ã
l
if ε is small enough. Then by the continuity of h there exists tε > 1, such that h(tε ) = 0, i.e., tε uε ∈ Na,β
.
Therefore, by hypothesis (f4 ) there exists µ2 > p, such that
tpε
∫
|x|
−ap
p
|∇uε | dx =
B
∫
β
|x| f (tε uε )tε uε
∫
β
|x| g(uε )uε dx
≥ tµε 2
B
∫
= ctµε 2 ε2
g(ψ)ψrN −1 H(εφ)drdφ,
B
Ã
i.e.,
−ap
p
|x|
|∇uε | dx
cε2 Ã g(ψ)ψrN −1 H(εφ)drdφ
)p
∫ (
ε2−p à ψ12 + r12 ψ22 2 r−ap r(ε−1)(N −p(a+1)) H(εφ)rN −1 drdφ
∫
=
cε2 Ã g(ψ)ψrN −1 H(εφ)drdφ
)p
∫ (
c1 ε2−p εN −l−1 Ã ψ12 + r12 ψ22 2 r−ap rN −1 drdφ
∫
≤
ε2 εN −l−1 Ã g(ψ)ψrN −1 drdφ
∫
tµε 2 −p
≤
B
∫
(23)
≤ c2 ε−p .
Furthermore, again by hypothesis (f4 ) we obtain the inequality
F (tv) ≥ tµ2 G(v)
(24)
R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120
119
∫v
for all t > 1 and v > 0, where G(v) := 0 g(t)dt. Hence, using the relations (23) and (24), we have
∫
∫
tp
−ap
p
β
|x|
|∇uε | dx −
|x| F (tε uε )dx
mla,β ≤ Ia,β (tε uε ) = ε
p B
B
∫
∫
tp
−ap
p
β
|x|
|∇uε | dx − tµε 2
|x| G(uε )dx
≤ ε
p B
B
)p
∫ (
−p2
1 2 2 −ap (ε−1)(N −p(a+1))
2−p
2
µ
−p
ψ1 + 2 ψ2 r
r
H(εφ)rN −1 drdφ
≤ c1 ε 2 ε
r
Ã
∫
−pµ2
G(ψ)rN −1 H(εφ)drdφ
− c2 ε µ2 −p ε2
(25)
Ã
≤ c3 ε
= c5
−p2
µ2 −p
−pµ2
ε2−p εN −l−1 + c4 ε µ2 −p ε2 εN −l−1
µ (p−1)+p
− 2 µ −p
+N −l
2
ε
(
= c5
N
β+N
)− µ2 (p−1)+p
+N −l
µ −p
2
≤ c6 β
µ2 (p−1)+p
−N +l
µ2 −p
. □
Finally, we will complete the proof of Theorem 1.2, i.e., we will prove that mla,β ̸= ma,β,r .
Completing the proof of Theorem 1.2. Applying the relation (2) present in the hypothesis (f4 ), we
obtain
β
µ2 (p−1)+p
−N +l
µ2 −p
<β
µ1 (p−1)+p
µ1 −p
for β large enough. Therefore, using Propositions 3.6 and 4.2, we can conclude that
mla,β < ma,β,r , as β → +∞. □
Acknowledgments
Olimpio Hiroshi Miyagaki was supported by INCTmat/MCT/Brazil, CNPq/Brazil Proc. 304015/2014-8
and Fapemig/Brazil CEX APQ-00063/15. Gilberto de Assis Pereira was supported by CNPq/Brazil. Bruno
Mendes Rodrigues was supported by Capes/DS.
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