Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 20, pp. 1–18.

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

HÉNON EQUATION WITH NOLINEARITIES INVOLVING

SOBOLEV CRITICAL GROWTH IN H1
0,rad(B1)

EUDES M. BARBOZA, OLIMPIO H. MIYAGAKI,

FÁBIO R. PEREIRA, CLÁUDIA R. SANTANA

Abstract. In this article we study the Hénon equation

−∆u = λ|x|µu+ |x|α|u|2
∗
α−2u in B1,

u = 0 on ∂B1,

where B1 is the ball centered at the origin of RN (N ≥ 3) and µ ≥ α ≥ 0.

Under appropriate hypotheses on the constant λ, we prove existence of at least
one radial solution using variational methods.

1. Introduction

In this article we search for a non-trivial radially symmetric solution to the
Hénon-type Dirichlet problem

−∆u = λ|x|µu+ |x|α|u|2
∗
α−2u in B1,

u = 0 on ∂B1,
(1.1)

where λ > 0, µ ≥ α ≥ 0, B1 is a unity ball centered at the origin of RN (N ≥ 3),

and 2∗α = 2(N+α)
N−2 .

When α = µ = 0, the pioneering work is due to Brézis and Nirenberg in [9],
where they obtained a λ1 and positive solutions when λ < λ1. We refer the reader
to the book [39] for a survey about this subject. Devillanova and Solimini [24]
proved multiplicity results for N ≥ 7, for all λ > 0. Then in [25], they comple-
mented the former result for N ≥ 4, but for λ ∈ (0, λ1). Clapp and Weth [20]
extended the above results for N ≥ 4, for all λ > 0, getting lower estimates for the
number of solutions. Chen, Shioji and Zou [18] obtained a ground state solution
and multiplicity results, and improved results in [20]. The existence is proved in
[15], for all λ > 0 and N ≥ 5, and when N = 4 only for λ 6= λk, where λk is
eigenvalue of (−∆). In [17] some multiplicity results were obtained for λ near λk.
These existence results were improved in [26]. For a version of these results in the
quasilinear see [21, 1].

When α, µ > 0, these problems are called Hénon type problems. Actually,
Hénon [28] introduced problem (1.1) with λ = 0, as a model of clusters of stars
for the case N = 1. Since then, many authors have worked with this type of

2010 Mathematics Subject Classification. 35J20, 35J25, 35B33, 35B34.
Key words and phrases. Hénon type equation; critical Sobolev growth; resonance;

noncompact variational problem.
c©2021 Texas State University.

Submitted October 2, 2019. Published March 29, 2021.

1



2 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

the equations from several points of view. The pioneering paper is due to Ni
[32]; he established the compact embedding H1

0,rad(B1) ⊂ Lp(B1, |x|α) for all p ∈
[1, 2∗α), where 2∗α = 2(N+α)

N−2 . This was used for obtaining radial solutions. Here

H1
0,rad(B1) = {u ∈ H1

0 (B1) : u is radial, that is, u(x) = u(|x|),∀x ∈ B1}. This

result was extended to more general quasilinear operators in [21]. In the case λ = 0,
Badiale and Serra [2] obtained multiplicity results for non-radial solutions (see [16]
for some extensions). For ground state profile (when the solutions that concentrate
at a boundary point of B1 as α→∞) and when the growth approaches to the usual
Sobolev critical exponent, see [10, 11, 13, 14, 30, 34, 38], and references therein.
For Hénon problems involving the usual Sobolev exponents we cite [31, 29, 35, 36]
and their references. Up to our knowledge, there are only a few works treating
problem (1.1) with λ 6= 0 involving the Sobolev critical exponent given by Ni, 2∗α.
Nonhomogeneous perturbations are studied in [3], when λ > 0 and smaller than
the first eigenvalue. While some concentration phenomena for linear perturbation
is studied in [27] when λ is small enough. Long and Yang [31] established the
existence of nontrivial solutions for (1.1) with µ = 0, when λ 6= λk, for all k, and
N ≥ 7. Also, they proved that (λk, 0) is a bifurcation point of problem (1.1), for
all k. The aim of this article is to extend above results, for instance, treating all λ
positive.

To establish our results, we need to know the spectrum of the problem

−∆u = λ|x|µu in B1;

u = 0 on ∂B1.
(1.2)

Note thatH1
0,rad(B1) is a Hilbert space, which is compactly embedded in Lp(B1, |x|µ),

for all p ∈ (1, 2∗µ) (see [32]). Arguing as in [22, 4], we can show that there exists a
sequence of eigenvalues for (1.2), with

λ∗1 ≤ λ∗2 ≤ λ∗3 ≤ · · · ≤ λ∗k ≤ . . . , λ∗k → +∞, as k →∞.
The eigenvalues are characterized by

λ∗1 = min
u∈H1

0,rad(B1)\{0}

∫
B1
|∇u|2dx∫

B1
|x|µ|u|2dx

, λ∗k+1 = min
u∈Pk+1\{0}

∫
B1
|∇u|2 dx∫

B1
|x|µ|u|2 dx

, (1.3)

where

Pk+1 =
{
u ∈ H1

0,rad(B1) : 〈u, ej〉 =

∫
B1

∇u∇ej dx = 0, j = 1, 2, . . . , k
}
, (1.4)

and ek denotes the eigenfunction associated with the eigenvalue λ∗k. Also from [22],
we know that e1 > 0, and that ej for j 6= 1 changes sign.

The results below follow from the linear theory, which are obtained by adapting
the ideas in [7] or [37, Appendix A]):

(1) each λ∗k has finite multiplicity,

(2) ek ∈ C0,σ(B1) for some σ ∈ (0, 1);
(3) the sequence {ek} is an orthonormal basis in L2(B1, |x|µ) and orthogonal

in H1
0,rad(B1).

For a fix k ∈ N we can assume λ∗k < λ∗k+1, otherwise we can assume that λ∗k has
multiplicity p ∈ N; that is,

λ∗k−1 < λ∗k = λ∗k+1 = . . . = λ∗k+p−1 < λ∗k+p,

and we denote λ∗k+p = λ∗k+1.



EJDE-2021/20 HÉNON EQUATION 3

The proofs of our results are based on variational methods. To ensure that the
considered minimax levels lie in a suitable range, we use approximating functions
that are constructed from Talenti functions (Hénon version). When we work with
nonlinearities involving Sobolov critical growth, it is common to follow the Brézis-
Nirenberg approach to estimate the minimax levels with the help of the Talenti
functions,

Uε(x) =
[N(N − 2)ε

ε+ |x|2
](N−2)/4

(1.5)

which are solutions of the problem

−∆u = |u|2
∗−2u in RN ;

u(x)→ 0 as |x| → ∞.

It is well-know that they yield the best Sobolev embedding constant constant for
H1(RN ) ⊂ L2∗(RN ), given by

S = inf
u∈H1

0 (B1),u 6=0

‖u‖2

‖u‖22∗
.

Using Uε one can prove that the minimax level of the functional associated with
problems with critical growth belongs to the interval where the Palais-Smale com-
pactness condition holds.

When searching for solutions to Hénon type equations in H1
0,rad(B1), we note

that the weight |x|α modifies the critical exponent, it becomes 2∗α ≥ 2∗ for α ≥ 0.
Consequently, we need to invoke a different family of functions adapted for the radial
context. More precisely, since we are searching for radial solutions for (1.1) with
critical growth, we let Sα be the best constant for the Sobolev-Hardy embedding

H1
0,rad(RN )→ L2∗α(RN , |x|α).

The constant is

Sα = inf
u∈H1

0,rad(B1), u 6=0

∫
RN |∇u|

2 dx( ∫
RN |x|α|u|2

∗
α dx

)2/2∗α (1.6)

which is achieved by the family of functions

uε(x) =
[(N + α)(N − 2)ε](N−2)/2(2+α)

(ε+ |x|2+α)(N−2)/(2+α)
(1.7)

defined for ε > 0. Indeed, these functions are minimizers of Sα in the set of radial
functions in the case α > −2. Furthermore, the uεs are the positive radial solutions
of

−∆u = |x|α|u|2
∗
α−2u in RN ;

u(x)→ 0 as |x| → ∞.
(1.8)

For details and more general results, see [3, 12, 19, 21, 32].

1.1. Statement of main results. We present our results in three theorems. The
first theorem deals with the non-trivial solution of problem (1.1) when λ > 0 and
N > 4 + µ. The possibility of resonance is also considered in this case. The
second theorem also concerns the non-trivial solution, when the working dimension
is 4 + µ; in this case we need to consider λ 6= λ∗j for j ∈ N = {1, 2, 3, . . . }. In
the third theorem considers non-trivial solutions when N < 4 + µ. To recover the



4 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

compactness of the functional associated with problem (1.1), we need λ large, with
λ 6= λ∗j .

Theorem 1.1. For 0 < λ < λ∗1 or λ∗k ≤ λ < λ∗k+1, problem (1.1) possesses a
non-trivial radial solution when

N >
µ− α

2
+ 2 + (2 + µ)

√
2. (1.9)

Theorem 1.2. For 0 < λ < λ∗1 or λ∗k < λ < λ∗k+1, problem (1.1) possesses a
non-trivial radial solution when N = 4 + µ.

Theorem 1.3. For λ > 0 sufficiently large and λ 6= λ∗j , for j ∈ N, problem (1.1)
possesses a non-trivial radial solution when N < 4 + µ.

Remark 1.4. Observe that (1.9) implies N > 4 + µ. In this sense, Theorem
1.1 provides a partial answer to the question about existence of nontrivial radial
solutions when N > 4 + µ.

In [3], it was proved that the non-trivial solution of (1.1) is positive when 0 <
λ < λ∗1.

This article is organized as follows. In Section 2, we introduce the variational
framework, prove the boundedness of Palais-Smale sequences of the functional as-
sociated with problem (1.1). Since we search for a radial solutions for a problem
with critical Sobolev growth nonlinearity, we show the minimax levels are bounded
by constants depending on N , α and Sα. In Section 3, we obtain the geometric
conditions on the functional for proving the existence of solutions to (1.1). In Sec-
tion 4, following [15], we obtain estimates for recovering the compactness of the
functional associated with problem (1.1). In Section 5, we prove our main results.

2. Variational formulation

Given a real Banach space E and a functional Φ of class C1 on E, by definition
Φ satisfies Palais-Smale condition at level c ∈ R (denoted (PS)c) if every sequence
(uj) in E such that

Φ(uj)→ c and Φ′(uj)→ 0 in E∗ (2.1)

has a convergent subsequence. Such a sequence is called a (PS) sequence (at level
c). We shall use the following version of a well-known critical-point theorem (see
[5]).

Theorem 2.1. Let H be a real Hilbert space and f ∈ C1(H,R) be a functional
satisfying the following assumptions:

(1) f(u) = f(−u), f(0) = 0 for any u ∈ H;
(2) there exists β > 0 such that f satisfies (PS)c for c ∈ (0, β);
(3) there exist two closed subspaces V,W ⊂ H and positive constants ρ, δ with

δ < β such that
(i) f(u) < β for any u ∈W ;
(ii) f(u) ≥ δ for any u ∈ V , ‖u‖ = ρ;

(iii) codimV <∞.

Then there exist at least m pairs of critical points, where

m = dim(V ∩W )− codim(V +W ).



EJDE-2021/20 HÉNON EQUATION 5

We consider H1
0,rad(B1), with the norm

‖u‖ =
(∫

B1

|∇u|2 dx
)1/2

.

The subspace of functions inB1 with weight |x|µ and µ ≥ 0 is denoted by Lz(B1, |x|µ),
and it is endowed the norm

‖u‖z,|x|µ =
(∫

B1

|x|µ|u|z dx
)1/z

.

For finding (weak) solutions of (1.1) we look for critical points of the functional
Jλ : H1

0,rad(B1)→ R defined as

Jλ(v) =
1

2

∫
B1

(|∇v|2 − λ|x|µv2) dx− 1

2∗α

∫
B1

|x|α|v|2
∗
α dx.

We do not apply the standard variational arguments because the embedding of
H1

0,rad(B1) in L2∗α(B1, |x|α) is not compact, and that the functional Jλ does not
satisfy the Palais-Smale condition. We need to adapt an idea introduced by Brézis
and Nirenberg [9] and Secchi [35]. This idea was used for the Talenti functions
(1.5) for proving that a functional associated with a problem with critical Sobolev
growth nonlinearity satisfies the PS-condition in the interval (0, SN/2/N).

Here, in the radial context for a Hénon type equation, we construct minimax
levels for the functional Jλ which lie in the interval(

0,
2 + α

2(N + α)
S(N+α)/(2+α)
α

)
.

For this purpose, we use that positive solutions (1.7) of (1.8) yield the constant Sα
in the embedding of H1

0,rad(RN ) in L2∗α(RN , |x|α).

2.1. Palais-Smale sequences. Recall that the proof of the Palais-Smale condition
for the functional associated with Problem (1.1) follows traditional methods. So
we present a brief proof for this condition.

Lemma 2.2. Let (um) ⊂ H1
0,rad(B1) be a (PS)c sequence of Jλ. Then (um) is

bounded in H1
0,rad(B1).

Proof. Let (um) ⊂ H1
0,rad(B1) be a (PS)c sequence, that is

Jλ(um) =
1

2
‖um‖2 −

λ

2
‖um‖22,|x|µ −

1

2∗α

∫
B1

|x|α|um|2
∗
α dx = c+ o(1) (2.2)

and

〈J ′λ(um), v〉 =

∫
B1

∇um∇v dx− λ
∫
B1

|x|µumv dx−
∫
B1

|x|α|um|2
∗
α−2umv dx

= o(1)‖v‖
(2.3)

for all v ∈ H1
0,rad(B1). From (2.2) and (2.3), it follows that

Jλ(um)− 1

2
〈J ′λ(um), um〉 =

2∗α − 2

2 · 2∗α

∫
B1

|x|α|um|2
∗
α dx

=c+ o(1) + o(1)‖um‖.
(2.4)



6 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

Considering 0 < λ < λ∗1, by the variational characterization of λ∗1, we have

〈J ′λ(um), um〉 ≥
(

1− λ

λ∗1

)
‖um‖2 −

∫
B1

|x|α|um|2
∗
α dx.

Hence by (2.4), we obtain

‖um‖2 ≤ C1 + C2‖um‖

and consequently (um) is a bounded sequence in H1
0,rad(B1).

Now we consider λ∗k < λ < λ∗k+1. It is convenient to decompose H1
0,rad(B1) into

the following subspaces,

H1
0,rad(B1) = Hk ⊕H⊥k , (2.5)

where Hk is finite dimensional defined by

Hk = [e1, . . . , ek]. (2.6)

For u in H1
0,rad(B1), let u = uk + u⊥, where uk ∈ Hk and u⊥ ∈ (Hk)⊥. We note

that ∫
B1

∇u∇uk dx− λ
∫
B1

|x|µuuk dx = ‖uk‖2 − λ‖uk‖22,|x|µ , (2.7)∫
B1

∇u∇u⊥ dx− λ
∫
B1

|x|µuu⊥ dx = ‖u⊥‖2 − λ‖u⊥‖22,|x|µ . (2.8)

By (2.3) and (2.8), we can see that

〈Jλ(um), u⊥m〉 = ‖u⊥m‖2 − λ‖u⊥m‖22,|x|µ −
∫
B1

|x|α|um|2
∗
α−2umu

⊥
m dx = o(1)‖u⊥m‖.

Then, from the variational characterization of λ∗k+1, the Holder and Young inequal-
ities, and (2.4), we obtain(

1− λ

λ∗k+1

)
‖u⊥m‖2

≤
∫
B1

|x|α|um|2
∗
α−2umu

⊥
m dx+ o(1)‖u⊥m‖

≤
(∫

B1

|x|α|um|2
∗
α dx

) 2∗α−1

2∗α
(∫

B1

|x|α|u⊥m|2
∗
α dx

) 1
2∗α

≤ ε
(∫

B1

|x|α|u⊥m|2
∗
α dx

)2/2∗α
+ cε

(∫
B1

|x|α|um|2
∗
α dx

) 2(2∗α−1)

2∗α + o(1)‖u⊥m‖

≤ ε‖u⊥m‖2 + cε

(∫
B1

|x|α|um|2
∗
α dx

) 2(2∗α−1)

2∗α + c‖u⊥m‖.

By (2.4) and [32, Compactness Lemma] which guarantees the compact embedding
of H1

0,rad(B1) in Lz(B1, |x|α) for 2 ≤ z < 2∗α, we have

‖u⊥m‖2 ≤ (c+ c‖um‖)
2(2∗α−1)

2∗α + c‖u⊥m‖. (2.9)

For ukm ∈ Hk, using the variational characterization of λ∗k, similar to (2.9), we
obtain

‖ukm‖2 ≤ (c+ c‖um‖)
2(2∗α−1)

2∗α + c‖ukm‖. (2.10)



EJDE-2021/20 HÉNON EQUATION 7

By summing the inequalities in (2.9) and (2.10), we have

‖um‖2 ≤ (C + C‖um‖)
2(2∗α−1)

2∗α + C‖um‖,
which proves the boundedness of the sequence (um) in H1

0,rad(B1) as desired.
Lastly, we consider λ = λ∗k for some k ∈ N. We use the decomposition

H1
0,rad(B1) = Hk−1 ⊕H⊥k ⊕ Eλ∗k , (2.11)

where Eλ∗k is the eigenspace associated with eigenvalue λ∗k. For the sequence (um)

in H1
0,rad(B1), we have

um = uk−1
m + u⊥m + wm = vm + wm,

where uk−1
m ∈ Hk−1, u⊥m ∈ (Hk)⊥, vm = uk−1

m +u⊥m and wm =
∑l
i=1 yi,mei,λ∗k ∈ Eλ∗k ,

where ei,λ∗k is an eigenfunction associated with λ∗k for 1 ≤ i ≤ l, l is the multiplicity
of λ∗k, and wm can be consider different from 0 for all m ∈ N. Note that ‖wm‖ ≤ ym,
where ym = lmax{|yi,m|; 1 ≤ i ≤ l}. Using arguments similar to those used in (2.9)
and (2.10), we conclude that

‖vm‖2 ≤ C(1 + ‖um‖)
2(2∗α−1)

2∗α + C‖vm‖. (2.12)

We can assume ‖um‖ ≥ 1 (if ‖um‖ ≤ 1, the sequence (um) is bounded inH1
0,rad(B1))

and, since ‖um‖ ≤ ‖vm‖+ ym, by (2.12), we obtain

‖vm‖2 ≤ C(‖vm‖+ ym)
2(2∗α−1)

2∗α + C‖vm‖. (2.13)

If ym is bounded, from (2.13), we have that (vm) is bounded in H1
0,rad(B1) and,

consequently, (um) is bounded in H1
0,rad(B1). Now let us assume ym → +∞. Using

(2.13), we have

‖vm
ym
‖2 ≤ C

[ (‖vm‖+ ym)
(2∗α−1)

2∗α

ym

]2
+

C

ym
‖vm
ym
‖

≤ C
[ 1

y
1− (2∗α−1)

2∗α
m

‖vm
ym
‖

(2∗α−1)

2∗α +
1

y
1− (2∗α−1)

2∗α
m

]2
+

C

ym
‖vm
ym
‖.

(2.14)

Thus, we obtain

‖vm
ym
‖2 ≤ C‖vm

ym
‖

2(2∗α−1)

2∗α + C‖vm
ym
‖+ C,

which implies the sequence { vmym } being bounded because
(2∗α−1)

2∗α
< 1, and, by (2.14),

‖ vmym ‖ → 0 as m→ 0.

Therefore, possibly up to a subsequence, vm/ym → 0 a.e. in B1 and strongly in
Lq(B1, |x|α), 1 ≤ q < 2∗α. Notice that

〈J ′λ(um),
wm
ym
〉 =

1

y2
m

(∫
B1

|∇wm|2 dx− λ
∫
B1

|x|µw2
m dx

)
−
∫
B1

|x|α|um|2
∗
α−1wm

ym
dx = o(1)

(2.15)

and since wm ∈ Eλ∗k , we have

〈J ′λ(um),
wm
ym
〉 = −

∫
B1

|x|α|um|2
∗
α−1wm

ym
dx = o(1). (2.16)



8 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

Thus, we have∫
B1

|x|α|um
ym
|2
∗
α−2um

ym
wm dx =

1

y
2∗α−1
m

∫
B1

|x|α|um|2
∗
α−2um

wm
ym

dx→ 0 (2.17)

as n → ∞. Note, since um = vm + wm, we have that um
ym
→ w0 in Lq(B1, |x|α)

for all 1 ≤ q < 2∗α and a.e. in B1 with w0 ∈ Eλ∗k \ {0}. So, by the Dominated
Convergence Theorem and using (2.17), it follows that∫

B1

|x|α|um
ym
|2
∗
α−2um

ym

wm
ym

dx→
∫
B1

|x|α|w0|2
∗
α dx = 0 (2.18)

which is a contradiction. So ym is bounded and, consequently, (um) is also bounded
in H1

0,rad(B1). �

We need to show that the minimax levels are below a suitable constant. For this
purpose, we need an estimate that allows us to simplify some calculations needed
ahead. Initially, we consider a Palais-Smale sequence (um); thus, by Lemma 2.2,
we may assume that (eventually passing to a subsequence)

um ⇀ u ∈ H1
0,rad(B1),

um → u ∈ Lp(B1, |x|α) for any p ∈ [1, 2∗α[,

um → u ∈ Lp(B1, |x|µ) for any p ∈ [1, 2∗α[, if µ ≥ α,
um → u a.e. in B1.

(2.19)

To check that u is a solution for (1.1), we need the following lemma.

Lemma 2.3. Let (um) be a (PS)c sequence in H1
0,rad(B1), with

c <
2 + α

2(N + α)
S(N+α)/(2+α)
α ,

and let vm = um − u. Then vm → 0 strongly in H1
0,rad(B1).

Proof. By Lemma 2.2, ‖um‖ is bounded, so from (2.19), u is a weak solution of
(1.1). Then, by (2.3) we have

‖u‖2 − λ‖u‖22,|x|µ −
∫
B1

|x|α|u|2
∗
α dx = 0. (2.20)

By the Brézis-Lieb Lemma [8], it follows that∫
B1

|x|α|um|2
∗
α dx =

∫
B1

|x|α|vm|2
∗
α dx+

∫
B1

|x|α|u|2
∗
α dx+ o(1). (2.21)

On the other hand, since H1
0,rad(B1) is a Hilbert Space, we obtain

‖um‖2 = ‖vm‖2 + ‖u‖2 + o(1). (2.22)

By (2.2), (2.21), and (2.22), as um → u in L2(B1, |x|µ), we obtain

c+ o(1) =Jλ(um)

=Jλ(u) +
1

2
‖vm‖2 −

λ

2
‖vm‖22,|x|µ −

1

2∗α

∫
B1

|x|α|vm|2
∗
α dx+ o(1)

=Jλ(u) +
1

2
‖vm‖2 −

1

2∗α

∫
B1

|x|α|vm|2
∗
α dx+ o(1).

(2.23)



EJDE-2021/20 HÉNON EQUATION 9

Since J ′λ(u) = 0 and ‖vm‖22,|x|µ = o(1), we conclude that

〈J ′λ(um), vm〉 = ‖vm‖2 −
∫
B1

|x|α|vm|2
∗
α dx+ o(1).

Then

‖vm‖2 =

∫
B1

|x|α|vm|2
∗
α dx+ o(1). (2.24)

Now, by (2.3) and taking um as test function, we note that∫
B1

|x|α|um|2
∗
α dx = ‖um‖2 − λ‖um‖22,µ + o(1).

So, as um → u in L2(B1, |x|µ) and using (2.22), we obtain

Jλ(um) =
1

2
(‖um‖2 − λ‖um‖22,|µ|)−

1

2∗α

∫
B1

|x|α|um|2
∗
α dx

=
1

2
(‖um‖2 − λ‖um‖22,|µ|)−

1

2∗α
(‖um‖2 − λ‖um‖22,µ + o(1))

=
2 + α

2(N + α)
(‖um‖2 − λ‖um‖22,|x|µ) + o(1)

=
2 + α

2(N + α)
(‖u‖2 − λ‖u‖22,|x|µ + ‖vm‖2) + o(1).

(2.25)

From (2.20), we conclude that

‖u‖2 − λ‖u‖22,|x|µ ≥ 0. (2.26)

Thus, by (2.25) and (2.26), we have

‖vm‖2 ≤
2(N + α)

2 + α
Jλ(um) + o(1).

By (2.2), since c < 2+α
2(N+α)S

(N+α)/(2+α)
α , for m sufficiently large we obtain

‖vm‖2 ≤ c+ o(1) < S(N+α)/(2+α)
α . (2.27)

From (1.6) and (2.24), we obtain

‖vm‖2 ≤ S
−2∗α/2
α ‖vm‖2

∗
α + o(1),

which implies

‖vm‖2(S2∗α/2 − ‖vm‖2
∗
α−2) ≤ o(1).

This and (2.27) imply that vm → 0 strongly in H1
0,rad(B1). �

3. Geometric conditions

Here we prove that Jλ satisfies the geometric condition of Theorem 2.1. Firstly,
given λ > 0, we define λ+ = min{λ∗j : λ < λ∗j} and set

H1 = ⊕[ej ]λ∗j≥λ+

H1
0,rad(B1)

H2 = [e1, . . . , ej ]λ∗j<λ+ . (3.1)

Lemma 3.1. There exist δ, ρ > 0 such that, for u ∈ H1,

Jλ(u) ≥ δ if ‖u‖ = ρ.



10 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

Proof. Let us take u ∈ H1, by the variational characterization of λ+ we obtain that

Jλ(u) ≥ 1

2

(
1− λ

λ+

)
‖u‖2 − C‖u‖2

∗
α ≥ δ > 0

when ‖u‖ = ρ with ρ > 0 small enough. �

4. Estimates of minimax levels

In this section, we obtain some estimates to show that the minimax levels are
below an appropriate constant in order to recover a similar compactness property
for the functional Jλ.

First, let r ∈ (0, 1) and Br = {x ∈ RN : |x| ≤ r}. We take ξr ∈ C∞0 (Br, [0, 1]), a
radial cut-off function such that ξr = 1 in Br/2 and |∇ξr| ≤ 4/r, and set urε(x) =
ξr(x)uε(x). In [3, Proof of Theorem 3.3] were obtained the following estimates of
Brézis-Nirenberg type [9, Lemma 1.2], which also can be found in [3, 21].

Lemma 4.1. Let K1,K2 and K3 be positive constants. For fixed r ∈ (0, 1) and
µ, α ≥ 0 and ε > 0 small enough, we have

(a) ‖urε‖2 = S
(N+α)/(2+α)
α +O

(
ε(N−2)/(2+α)

)
;

(b) ‖urε‖
2∗α
2∗α,|x|α

= S
(N+α)/(2+α)
α +O

(
ε(N+α)/(2+α)

)
;

(c)

‖urε‖22,|x|µ =


K1ε

(2+µ)/(2+α) if N > 4 + µ;

K1ε
(2+µ)/(2+α)| log ε|+O

(
ε(2+µ)/(2+α)

)
if N = 4 + µ;

K1ε
(N−2)/(2+α) if N < 4 + µ;

(d) ‖urε‖1,|x|µ ≤ K2ε
(N−2)/[2(2+α)];

(e) ‖urε‖
2∗α−1

2∗α−1,|x|α ≤ K3ε
(N−2)/[2(2+α)].

Now we shall prove some main technical lemmas. First of all, we define

W (ε, r) = {u ∈ H1
0,rad(B1);u = u− + turε , u

− ∈ H2, t ∈ R}.

Remark 4.2. Since uε is solution for (1.8), urε 6∈ [e1, e2, . . . , ek] for any k ∈ N.
Thus, W (ε, r) 6= H2.

Lemma 4.3. If u ∈W (ε, r), then for ε > 0 sufficiently small

‖u‖2
∗
α

2∗α,|x|α
≥ ‖turε‖

2∗α
2∗α,|x|α

− Ct2
∗
αε(N−2)(N+α)/[(N+2α+2)(2+α)] (4.1)

for any t ∈ R.

Proof. Note that from

‖u‖2
∗
α

2∗α,|x|α
= 2∗α

∫
B1

|x|α dx

∫ u

0

|s|2
∗
α−2sds, (4.2)

and the Mean Value Theorem, we obtain

‖u‖2
∗
α

2∗α,|x|α
− ‖turε‖

2∗α
2∗α,|x|α

− ‖u−‖2
∗
α

2∗α,|x|α

= 2∗α

∫ 1

0

ds

∫
B1

|x|α[|turε + su−|2
∗
α−2(turε + su−)− |su−|2

∗
α−2su−]u− dx

= 2∗α(2∗α − 1)

∫ 1

0

ds

∫
B1

|x|α|turε + τsu−|2
∗
α−2turε · u− dx

(4.3)



EJDE-2021/20 HÉNON EQUATION 11

where τ = τ(x) is a measurable function such that 0 < τ(x) < 1.
Using (4.3) and since u− ∈ H2, which is a finite-dimension subspace, we obtain∣∣‖u‖2∗α2∗α,|x|α − ‖turε‖2∗α2∗α,|x|α − ‖u−‖2∗α2∗α,|x|α ∣∣

≤ C
∫ 1

0

ds

∫
B1

|x|α(|turε |2
∗
α−1|u−|+ |u−|2

∗
α−1|turε |) dx

≤ C‖turε‖
2∗α−1

2∗α−1,|x|α‖u
−‖∞ + ‖u−‖2

∗
α−1

∞,|x|α‖tu
r
ε‖1

≤ C‖turε‖
2∗α−1

2∗α−1,|x|α‖u
−‖2 + ‖u−‖2

∗
α−1

2∗α,|x|α
‖turε‖1,

(4.4)

where C is positive constant. From (4.4), the Young inequality and the items (d)
and (e) of Lemma 4.1, we have that∣∣‖u‖2∗α2∗α,|x|α − ‖turε‖2∗α2∗α,|x|α − ‖u−‖2∗α2∗α,|x|α ∣∣
≤ Ct2

∗
α−1ε(N−2)/(2(2+α))‖u−‖2 +

N + 2 + 2α

2(N + α)
‖u−‖2

∗
α

2∗α,|x|α
+ Ct2

∗
αε(N+α)/(2+α).

Finally, again by the Young inequality, we have∣∣‖u‖2∗α2∗α,|x|α − ‖turε‖2∗α2∗α,|x|α − ‖u−‖2∗α2∗α,|x|α ∣∣
≤ Ct2

∗
α−1ε

(N−2)
(2(2+α)) ‖u−‖2∗α,|x|α +

N + 2 + 2α

2(N + α)
‖u−‖2

∗
α

2∗α,|x|α
+ Ct2

∗
αε

(N+α)
(2+α)

≤ Ct2
∗
αε

(N−2)(N+α)
[(N+2α+2)(2+a)] +

1

2∗α
‖u−‖2

∗
α

2∗α,|x|α
+
N + 2 + 2α

2(N + α)
‖u−‖2

∗
α

2∗α,|x|α
+ Ct2

∗
αε

(N+α)
(2+α)

= Ct2
∗
αε

(N−2)(N+α)
[(N+2α+2)(2+α)] + ‖u−‖2

∗
α

2∗α,|x|α
+ Ct2

∗
αε

(N+α)
(2+α)

≤ Ct2
∗
αε

(N−2)(N+α)
[(N+2α+2)(2+α)] + ‖u−‖2

∗
α

2∗α,|x|α
.

for ε > 0 small enough. The proof is complete. �

Lemma 4.4. For ε > 0 sufficiently small, we have

‖urε‖2 − λ‖urε‖22,|x|µ
‖urε‖22∗α,|x|α

=


Sα − Cε(2+µ)/(2+α) if N > 4 + µ;

Sα − Cε(2+µ)/(2+α)| log(ε)|+O(ε(2+µ)/(2+α)) if N = 4 + µ;

Sα + ε(N−2)/(2+α)(O(1)− λC) if N < 4 + µ.

(4.5)

The statement of the lemma above is obtained from (a)–(c) in Lemma 4.1.
Now we separate our study into three cases: non-resonant case assuming (1.9),

and consequently, N > 4 + µ, or N = 4 + µ; resonant case when (1.9) holds; and
non-resonant case with N < 4 + µ. This separation occurs because to prove the
(PS)c condition for c below an appropriate constant when λ = λj for some j ∈ N,
we need to have N > 4 + µ. When N < 4 + µ, it is crucial to assume in addition
that λ is sufficiently large to prove the (PS)c condition.

4.1. Non-resonant case with N ≥ 4 +µ. Initially, we consider the non-resonant
case and we obtain the following results.



12 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

Lemma 4.5. Assume (1.9), for ε sufficiently small and positive. If λ 6= λ∗j , for
every j ∈ N, then

sup
W (ε,r)

Jλ(u) <
(2 + α)

2(N + α)
S(N+α)/(2+α)
α . (4.6)

Proof. Note that for fixed u ∈ H1
0,rad(B1) with u 6= 0, we obtain

sup
t
Jλ(tu) =

(2 + α)

2(N + α)

(‖u‖2 − λ‖u‖22,|x|µ
‖u‖22∗α,|x|α

)(N+α)/(2+α)

. (4.7)

Since

sup{Jλ(u) : u ∈W (ε) \ {0}}

= sup
{
Jλ(‖u‖2∗α,|x|α

u

‖u‖2∗α,|x|α
) : u ∈W (ε, r) \ {0}

}
≤ sup{Jλ(tu) : u ∈W (ε, r) \ {0} with ‖u‖2∗

α,|x|α
= 1 and t ∈ R},

to show that (4.6) is true, we need to estimate

sup
u∈W (ε,r), ‖u‖2∗α,|x|α=1

{
‖u‖2 − λ‖u‖22,|x|µ

}
. (4.8)

Let u = u− + turε ∈ W (ε, r) with ‖u‖2∗α,|x|α = 1. By (4.1) and item (b) of Lemma
4.1, for ε small enough, we have

1 = ‖u‖2
∗
α

2∗α,|x|α

≥ ‖turε‖
2∗α
2∗α,|x|α

− Ct2
∗
αε(N−2)(N+α)/(N+2α+2)(2+α)

= t2
∗
α

(
S(N+α)/(2+α)
α +O

(
ε(N−2)/(2+α)

))
− Ct2

∗
αε(N−2)(N+α)/(N+2α+2)(2+α)

= t2
∗
α

(
S(N+α)/(2+α)
α +O

(
ε(N−2)(N+α)/(N+2α+2)(2+α)

))
.

Thus, we can conclude that t is bounded for small positive ε. From item (e) in
Lemma 4.1, the variational characterization of λ∗j and Green’s Theorem, we obtain

‖u‖2 − λ‖u‖22,|x|µ
≤ ‖turε‖2 − λ‖turε‖22,|x|µ + ‖u−‖2 − λ‖u−‖22,|x|µ

+ 2

∫
B1

{|turε | |∆u−|+ λ|x|µ|u−||turε |}dx

≤ ‖turε‖2 − λ‖turε‖22,|x|µ + ‖u−‖2 − λ‖u−‖22,|x|µ + C{‖turε‖1 ‖∆u−‖∞
+ λ‖u−‖∞‖turε‖1,|x|µ}

≤ ‖turε‖2 − λ‖turε‖22,|x|µ + ‖u−‖2 − λ‖u−‖22,|x|µ + C‖u−‖2ε(N−2)/[2(2+α)]

≤
‖turε‖2 − λ‖turε‖22,|x|µ

‖turε‖22∗α,|x|α
‖turε‖22∗α,|x|α + (λ− λ)‖u−‖22,|x|µ

+ C‖u−‖2,|x|µε(N−2)/[2(2+α)],

(4.9)

where λ = max{λ∗j : λ∗j < λ}.



EJDE-2021/20 HÉNON EQUATION 13

Now we define A(u−, ε, c) = (λ− λ)‖u−‖22,|x|µ +C‖u−‖2,|x|µε(N−2)/[2(2+α)]. No-

tice that

A(u−, ε, c) ≤ 0 or A(u−, ε, c) ≤ c2

λ− λ
ε(N−2)/(2+α). (4.10)

On the other hand by (4.1) and the boundedness of t, we obtain

‖turε‖22∗α,|x|α ≤
(
1 + Cε(N−2)(N+α)/[(N+2α+2)(2+α)]

)2/2∗α
≤ 1 + Cε(N−2)(N+α)/[(N+2α+2)(2+α)].

(4.11)

From (1.9), we obtain N > 4+µ, then using (4.5), (4.9), (4.10) and (4.11), we have

‖u‖2 − λ‖u‖22,|x|µ

≤
(
Sα − Cε(2+µ)/(2+α)

)(
1 + Cε[(N−2)(N+α)]/[(N+2α+2)(2+α)]

)
+A(u−, ε, c).

(4.12)
By (1.9), we also conclude that

(N − 2)(N + α)

(N + 2α+ 2)(2 + α)
>

2 + µ

2 + α
.

Thus, ‖u‖2 − λ‖u‖22,|x|µ < Sα for ε positive and small enough. �

Lemma 4.6. For ε > 0 sufficiently small and N = 4 + µ, if λ 6= λ∗j , for every
j ∈ N, then

sup
W (ε,r)

Jλ(u) <
(2 + α)

2(N + α)
S(N+α)/(2+α)
α . (4.13)

Proof. When N = 4+µ, as for (4.12), from (4.5), (4.9), (4.10) and (4.11), we obtain

‖u‖2 − λ‖u‖22,|x|µ ≤
(
Sα − Cε(2+µ)/(2+α)| log(ε)|+O(ε(2+µ)/(2+α))

)
×
(

1 + Cε[(2+µ+α)(4+µ+α)]/[(6+µ+2α)(2+α)]
)

+A(u−, ε, c).

Because of the behavior of | log(ε)| near zero, for ε small enough we conclude the
result. �

4.2. Resonant case with N > 4 + µ. Now we consider, λ = λ∗j for some j ∈ N.
We will find estimates which will help us in obtaining a result similar to Lemma
4.5 for the resonant case when (1.9) is satisfied.

First, we denote by Pj the projector on the eigenspace corresponding to λ∗j and
set

ũrε = urε − Pjurε . (4.14)

Thus, by item (d) in Lemma 4.1, we have

‖Pjurε‖22,|x|µ =
∑
k

(∫
B1

|x|µekurε dx
)2

≤ C‖urε‖21,|x|µ ≤ Cε
(N−2)/(2+α). (4.15)

Consequently, as Pju
r
ε is in a finite dimensional space, we obtain

‖Pjurε‖∞,|x|µ ≤ Cε(N−2)/2[(2+α)]. (4.16)

Furthermore,∣∣‖ũrε‖2∗α2∗α,|x|α − ‖urε‖2∗α2∗α,|x|α ∣∣



14 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

= 2∗α
∣∣ ∫ 1

0

ds

∫
B1

|x|α|urε − sPjurε |2
∗
α−2(urε − sPjurε)Pjurε dx

∣∣
≤ 2∗α · 22∗α−1

∫ 1

0

ds

∫
B1

|x|α
{
|urε |2

∗
α−1 + s2∗α−1|Pjurε |2

∗
α−1
}
|Pjurε | dx

≤ C
{
‖urε‖

2∗α−1

2∗α−1,|x|α‖Pju
r
ε‖∞,|x|µ + ‖Pjurε‖

2∗α
2,|x|α

}
.

Then from item (e) in Lemma 4.1, (4.15) and (4.16), we obtain∣∣‖ũrε‖2∗α2∗α,|x|α − ‖urε‖2∗α2∗α,|x|α ∣∣ ≤ Cε(N−2)/(2+α). (4.17)

By item (e) in Lemma 4.1 and (4.16), we notice that

‖ũrε‖
2∗α−1

2∗α−1,|x|α = ‖urε − Pjurε‖
2∗α−1

2∗α−1,|x|α

≤ C{‖urε‖
2∗α−1

2∗α−1,|x|α + ‖Pjurε‖
2∗α−1

2∗α−1,|x|α}

≤ Cε(N−2)/[2(2+α)].

(4.18)

As for (4.18), using item (d) in Lemma 4.1 and (4.16), we obtain

‖ũrε‖1,|x|µ ≤ Cε(N−2)/[2(2+α)] . (4.19)

Based on these estimates, we can conclude the following lemma.

Lemma 4.7. For ε sufficiently small and positive, we have

‖ũrε‖2 − λ‖ũrε‖22,|x|µ
‖ũrε‖22∗α,|x|α

= Sα − Cε(2+µ)/(2+α) if N > 4 + µ. (4.20)

The proof of the above lemma follows from (4.17), (4.18) and (4.19), and argu-
ments similar to those in Lemma 4.4. Now, we define

W̃ (ε) = {u ∈ H1
0,rad(B1) : u = u− + tũrε , u

− ∈ H2, t ∈ R}.

Arguments analogous to those used in the Lemma 4.5, guarantee the following
result.

Lemma 4.8. Suppose (1.9) and λ = λ∗j , for some j ∈ N. Then, for ε positive and
sufficiently small,

sup
W̃ (ε)

Jλ(u) <
(2 + α)

2(N + α)
S(N+α)/(2+α)
α . (4.21)

4.3. Non resonant case with N < 4 + µ. In this case, to conclude a similar
result to Lemma 4.5, we need another condition on λ. More precisely, we should
have λ sufficiently large to guarantee that the minimax levels are below a suitable
constant.

Lemma 4.9. Suppose N < 4 + µ and λ 6= λ∗j , for some j ∈ N. Then, for ε > 0
sufficiently small and λ large enough,

sup
W (ε,r)

Jλ(u) <
(2 + α)

2(N + α)
S(N+α)/(2+α)
α . (4.22)

Proof. As in Lemma 4.5, we need to show that

‖urε‖2 − λ‖urε‖22,|x|µ < Sα, (4.23)



EJDE-2021/20 HÉNON EQUATION 15

when λ 6= λ∗j for all j ∈ N. Thus, following the same steps as in Lemma 4.5, and
using (4.5) we obtain

‖u‖2 − λ‖u‖22,|x|µ ≤
(
Sα + ε(N−2)/(2+α)(O(1)− λC)

)
×
(

1 + Cε[(N−2)(N+α)]/[(2+α)(N+2α+2]
)

+A(u−, ε, C).

Therefore, for ε positive and small enough, and λ sufficiently large, we obtain
(4.23). �

5. Proof of main results

It is clear that Jλ ∈ C1(H1
0,rad(B1),R) and complies with condition (f1) of

Theorem 2.1. Then Lemma 2.3 ensures that (2) in Theorem 2.1 is satisfied with

β = (2+α)
2(N+α)S

(N+α)/(2+α))
α .

If 0 < λ 6= λ∗j for all j ∈ N, we set V = H1 and W = W (ε, r) with ε small
enough to satisfy Lemma 4.5 for N > 4 + µ, when (1.9) is satisfied, or Lemma 4.6
for N = 4 + µ. Then (3)(iii) in Theorem 2.1 holds in both cases. Thus, (3)(i)) and
(3)(ii)) are satisfied by Lemmas 3.1, 4.5 and 4.6, respectively. Since dim(V ∩W ) = 1
and V +W = H1

0,rad(B1), from Theorem 2.1, it follows that (1.1) has at least one
non trivial solution.

If 0 < λ = λ∗j for some j ∈ N and N > 4 + µ, when (1.9) is true, we conclude

this result repeating the above arguments using W = W̃ (ε) and the Lemma 4.8 and
3.1.

For N < 4+µ, following the same steps as in the two previous cases, Lemmas 4.9
and 3.1 with H1 = H1

0,rad(B1), we obtain the conclusion by applying Ambrosetti-

Rabinowitz Mountain Pass Theorem [39]. Recall that there is a function e ∈ H1

such that Jλ(e) ≤ 0. By standard arguments and the maximum principle, we can
show the solution is positive. This completes the proof.

Remark 5.1. We know that

J ′λ(v)w = 0, ∀w ∈ H1
0,rad(B1), (5.1)

and v is a critical point of the functional Jλ restricted to the space H1
0,rad(B1).

Now, we follow the ideas of [6, 23, 33]. Since H1
0,rad(B1) is a closed subspace of

H1
0 (B1), we can write

H1
0 (B1) = H1

0,rad(B1)⊕H1
0,rad(B1)⊥,

where ⊥ denotes the orthogonal complement of the space. Therefore, for each
w ∈ H1

0 (B1), there exist ϑ ∈ H1
0,rad(B1) and ϑ⊥ ∈ H1

0,rad(B1)⊥ such that

w = ϑ+ ϑ⊥. (5.2)

Since H1
0,rad(B1) is a Hilbert space and J ′λ(v) ∈ H1

0,rad(B1)∗, from the Riesz

Representation Theorem there exists z ∈ H1
0,rad(B1) such that

J ′λ(v)w =

∫
B1

∇z · ∇w dx, for all w ∈ H1
0,rad(B1).

Thus, J ′λ(v) ≈ z, as z ∈ H1
0,rad(B1) and ϑ⊥ ∈ H1

0,rad(B1)⊥, we have

J ′λ(v)ϑ⊥ = 0. (5.3)



16 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

From (5.1), (5.2) and (5.3), for each w ∈ H1
0 (B1), we obtain

J ′λ(v)w = J ′λ(v)ϑ+ J ′λ(v)ϑ⊥ = 0.

This implies that v is a critical point of the functional Jλ in H1
0 (B1) and conse-

quently v is a weak solution for problem (1.1).

Acknowledgments. O.H. Miyagaki was supported by grant 2019/24901-3 from
the São Paulo Research Foundation (FAPESP), and by grant 307061/2018-3 from
the CNPq/Brazil This article was written while C. R. Santana was on Postdoctoral
stage in the Department of Mathematics of the Federal University of Juiz de Fora,
whose hospitality she gratefully acknowledges. She would also like to express her
gratitude to Professor Olimpio H. Miyagaki. The authors would like to thank the
anonymous referees for their suggestions, which improved this article.

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Eudes M. Barboza

Departamento de Matemática, Universidade Federal Rural de Pernambuco, 50740-560,

Recife - PE, Brasil
Email address: eudes.barboza@ufrpe.br

Olimpio H. Miyagaki
Department of Mathematics, Universidade Federal de São Carlos, 13565-905, São Car-

los - SP, Brazil

Email address: olimpio@ufscar.br, ohmiyagaki@gmail.com

Fábio R. Pereira

Departamento de Matemática, Universidade Federal de Juiz de Fora, 36036-330 - Juiz

de Fora - MG, Brazil
Email address: fabio.pereira@ufjf.edu.br



18 E. M. BARBOZA, O. H. MIYAGAKI, F. R. PEREIRA, C. R. SANTANA EJDE-2021/20

Cláudia R. Santana

Departamento de Ciências Exatas e Tecnológicas, Universidade Estadual de Santa Cruz,

45662-900 Ilhéus - BA, Brazil
Email address: santana@uesc.br


	1. Introduction
	1.1. Statement of main results

	2. Variational formulation
	2.1. Palais-Smale sequences

	3. Geometric conditions
	4. Estimates of minimax levels
	4.1. Non-resonant case with N4+
	4.2. Resonant case with N>4+
	4.3. Non resonant case with N< 4+.

	5. Proof of main results
	Acknowledgments

	References