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

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

H ´ENON EQUATION WITH NOLINEARITIES INVOLVING SOBOLEV CRITICAL GROWTH IN H_0,rad¹ (B1)

EUDES M. BARBOZA, OLIMPIO H. MIYAGAKI, F ´ABIO R. PEREIRA, CL ´AUDIA R. SANTANA

Abstract. In this article we study the H´enon equation

−∆u=λ|x|^µu+|x|^α|u|^2∗α−2u inB1, u= 0 on∂B1,

whereB1 is the ball centered at the origin of R^N (N ≥3) andµ≥α≥0.

Under appropriate hypotheses on the constantλ, 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 H´enon-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 R^N (N ≥3), and 2^∗_α= ^2(N_N−2^+α).

When α = µ = 0, the pioneering work is due to Br´ezis and Nirenberg in [9], where they obtained aλ1 and positive solutions whenλ < 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 forN ≥4, for allλ >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 forλnear 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 H´enon type problems. Actually, H´enon [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

2010Mathematics Subject Classification. 35J20, 35J25, 35B33, 35B34.

Key words and phrases. H´enon type equation; critical Sobolev growth; resonance;

noncompact variational problem.

c

2021 Texas State University.

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

1

the equations from several points of view. The pioneering paper is due to Ni [32]; he established the compact embeddingH_0,rad¹ (B1)⊂L^p(B1,|x|^α) for allp∈ [1,2^∗_α), where 2^∗_α = ^2(N+α)_N−2 . This was used for obtaining radial solutions. Here H_0,rad¹ (B₁) = {u ∈ H₀¹(B₁) : uis radial, that is,u(x) = u(|x|),∀x ∈ B₁}. This result was extended to more general quasilinear operators in [21]. In the caseλ= 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 ofB₁asα→ ∞) and when the growth approaches to the usual Sobolev critical exponent, see [10, 11, 13, 14, 30, 34, 38], and references therein.

For H´enon 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) withλ6= 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) withµ= 0, when λ6=λk, for all k, and N ≥7. Also, they proved that (λk,0) is a bifurcation point of problem (1.1), for allk. The aim of this article is to extend above results, for instance, treating allλ positive.

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

−∆u=λ|x|^µu inB1;

u= 0 on∂B1. (1.2)

Note thatH_0,rad¹ (B₁) is a Hilbert space, which is compactly embedded inL^p(B₁,|x|^µ), for allp∈(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 →+∞, ask→ ∞.

The eigenvalues are characterized by λ^∗₁= min

u∈H_0,rad¹ (B₁)\{0}

R

B1|∇u|²dx R

B1|x|^µ|u|²dx, λ^∗_k+1= min

u∈Pk+1\{0}

R

B1|∇u|²dx R

B1|x|^µ|u|²dx, (1.3) where

P^k+1=

u∈H_0,rad¹ (B1) :hu, eji= Z

B1

∇u∇ejdx= 0, j= 1,2, . . . , k , (1.4) ande_k denotes the eigenfunction associated with the eigenvalueλ^∗_k. Also from [22], we know thate₁>0, and thate_j forj6= 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) e_k∈C^0,σ(B₁) for someσ∈(0,1);

(3) the sequence {e_k} is an orthonormal basis in L²(B₁,|x|^µ) and orthogonal inH_0,rad¹ (B1).

For a fix k∈ Nwe can assumeλ^∗_k < λ^∗_k+1, otherwise we can assume that λ^∗_k has multiplicityp∈N; that is,

λ^∗_k−1< λ^∗_k =λ^∗_k+1=. . .=λ^∗_k+p−1< λ^∗_k+p, and we denoteλ^∗_k+p=λ^∗_k+1.

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 (H´enon version). When we work with nonlinearities involving Sobolov critical growth, it is common to follow the Br´ezis- Nirenberg approach to estimate the minimax levels with the help of the Talenti functions,

U(x) =hN(N−2) +|x|²

i(N−2)/4

(1.5) which are solutions of the problem

−∆u=|u|^2∗−2u inR^N; u(x)→0 as |x| → ∞.

It is well-know that they yield the best Sobolev embedding constant constant for H¹(R^N)⊂L^2∗(R^N), given by

S= inf

u∈H₀¹(B1),u6=0

kuk² kuk²₂∗

.

Using U 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 H´enon type equations in H_0,rad¹ (B1), we note that the weight|x|^α modifies the critical exponent, it becomes 2^∗_α≥2^∗ forα≥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 letSα be the best constant for the Sobolev-Hardy embedding

H_0,rad¹ (R^N)→L^2∗α(R^N,|x|^α).

The constant is

Sα= inf

u∈H_0,rad¹ (B1), u6=0

R

R^N|∇u|²dx R

R^N|x|^α|u|^2∗αdx^2/2∗α

(1.6) which is achieved by the family of functions

u(x) = [(N+α)(N−2)]^{(N−2)/2(2+α)}

(+|x|^2+α)(N−2)/(2+α) (1.7)

defined for >0. Indeed, these functions are minimizers of Sα in the set of radial functions in the caseα >−2. Furthermore, theus are the positive radial solutions of

−∆u=|x|^α|u|^2∗α−2u inR^N;

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) whenλ >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 whenN <4 +µ. To recover the

compactness of the functional associated with problem (1.1), we needλlarge, with λ6=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. Forλ >0 sufficiently large andλ6=λ^∗_j, for j ∈N, problem (1.1) possesses a non-trivial radial solution whenN <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 whenN >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 S_α. 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 spaceE and a functional Φ of classC¹onE, by definition Φ satisfies Palais-Smale condition at levelc∈R(denoted (P S)c) if every sequence (uj) inE such that

Φ(u_j)→c and Φ⁰(u_j)→0 inE^∗ (2.1) has a convergent subsequence. Such a sequence is called a (P S) 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) = 0for any u∈H;

(2) there existsβ >0 such that f satisfies(P S)c forc∈(0, β);

(3) there exist two closed subspacesV, W ⊂H and positive constantsρ, δ with δ < β such that

(i) f(u)< β for any u∈W;

(ii) f(u)≥δfor any u∈V,kuk=ρ;

(iii) codimV <∞.

Then there exist at leastmpairs of critical points, where m= dim(V ∩W)−codim(V +W).

We considerH_0,rad¹ (B1), with the norm kuk=Z

B1

|∇u|²dx1/2

.

The subspace of functions inB1with weight|x|^µandµ≥0 is denoted byL^z(B1,|x|^µ), and it is endowed the norm

kuk_z,|x|µ =Z

B₁

|x|^µ|u|^zdx^1/z .

For finding (weak) solutions of (1.1) we look for critical points of the functional Jλ:H_0,rad¹ (B1)→Rdefined as

Jλ(v) = 1 2 Z

B1

(|∇v|²−λ|x|^µv²) dx− 1 2^∗_α

Z

B1

|x|^α|v|^2∗αdx.

We do not apply the standard variational arguments because the embedding of H_0,rad¹ (B1) in L^2∗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 Br´ezis 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, S^N/2/N).

Here, in the radial context for a H´enon type equation, we construct minimax levels for the functionalJ_λ 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 constantSα

in the embedding ofH_0,rad¹ (R^N) inL^2∗α(R^N,|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) ⊂ H_0,rad¹ (B1) be a (P S)c sequence of Jλ. Then (um) is bounded inH_0,rad¹ (B1).

Proof. Let (um)⊂H_0,rad¹ (B1) be a (P S)c sequence, that is Jλ(um) =1

2kumk²−λ

2kumk²_2,|x|µ− 1 2^∗_α

Z

B₁

|x|^α|um|^2∗αdx=c+o(1) (2.2) and

hJ_λ⁰(um), vi= Z

B₁

∇um∇vdx−λ Z

B₁

|x|^µumvdx− Z

B₁

|x|^α|um|^2∗α−2umvdx

=o(1)kvk

(2.3) for allv∈H_0,rad¹ (B1). From (2.2) and (2.3), it follows that

Jλ(um)−1

2hJ_λ⁰(um), umi=2^∗_α−2 2·2^∗_α

Z

B1

|x|^α|um|^2∗αdx

=c+o(1) +o(1)ku_mk.

(2.4)

Considering 0< λ < λ^∗₁, by the variational characterization ofλ^∗₁, we have hJ_λ⁰(u_m), u_mi ≥

1− λ λ^∗₁

ku_mk²− Z

B1

|x|^α|u_m|^2∗αdx.

Hence by (2.4), we obtain

kumk²≤C1+C2kumk

and consequently (um) is a bounded sequence inH_0,rad¹ (B1).

Now we considerλ^∗_k < λ < λ^∗_k+1. It is convenient to decomposeH_0,rad¹ (B1) into the following subspaces,

H_0,rad¹ (B₁) =H_k⊕H_k^⊥, (2.5) whereHk is finite dimensional defined by

Hk= [e1, . . . , ek]. (2.6) ForuinH_0,rad¹ (B₁), letu=u^k+u^⊥, whereu^k ∈H_k andu^⊥∈(H_k)^⊥. We note that

Z

B1

∇u∇u^kdx−λ Z

B1

|x|^µuu^kdx=ku^kk²−λku^kk²_2,|x|µ, (2.7) Z

B₁

∇u∇u^⊥dx−λ Z

B₁

|x|^µuu^⊥dx=ku^⊥k²−λku^⊥k²_2,|x|µ. (2.8) By (2.3) and (2.8), we can see that

hJλ(um), u^⊥_mi=ku^⊥_mk²−λku^⊥_mk²_2,|x|µ− Z

B₁

|x|^α|um|^2∗α−2umu^⊥_mdx=o(1)ku^⊥_mk.

Then, from the variational characterization ofλ^∗_k+1, the Holder and Young inequal- ities, and (2.4), we obtain

1− λ

λ^∗_k+1

ku^⊥_mk²

≤ Z

B₁

|x|^α|um|^2∗α−2umu^⊥_mdx+o(1)ku^⊥_mk

≤Z

B1

|x|^α|um|^2∗αdx

2∗ α−1

2∗ α Z

B1

|x|^α|u^⊥_m|^2∗αdx₂¹∗ α

≤Z

B₁

|x|^α|u^⊥_m|^2∗αdx^2/2∗α

+cZ

B₁

|x|^α|um|^2∗αdx

2(2∗ α−1) 2∗

α +o(1)ku^⊥_mk

≤ku^⊥_mk²+c

Z

B1

|x|^α|um|^2∗αdx

2(2∗ α−1) 2∗

α +cku^⊥_mk.

By (2.4) and [32, Compactness Lemma] which guarantees the compact embedding ofH_0,rad¹ (B₁) inL^z(B₁,|x|^α) for 2≤z <2^∗_α, we have

ku^⊥_mk²≤(c+ckumk)

2(2∗ α−1) 2∗

α +cku^⊥_mk. (2.9)

For u^k_m ∈ H_k, using the variational characterization of λ^∗_k, similar to (2.9), we obtain

ku^k_mk²≤(c+ckumk)

2(2∗ α−1) 2∗

α +cku^k_mk. (2.10)

By summing the inequalities in (2.9) and (2.10), we have kumk²≤(C+Ckumk)

2(2∗ α−1) 2∗

α +Ckumk,

which proves the boundedness of the sequence (um) inH_0,rad¹ (B1) as desired.

Lastly, we considerλ=λ^∗_k for some k∈N. We use the decomposition

H_0,rad¹ (B1) =H_k−1⊕H_k^⊥⊕Eλ^∗_k, (2.11) whereEλ^∗_k is the eigenspace associated with eigenvalueλ^∗_k. For the sequence (um) inH_0,rad¹ (B₁), we have

u_m=u^k−1_m +u^⊥_m+w_m=v_m+w_m, whereu^k−1_m ∈Hk−1,u^⊥_m∈(Hk)^⊥,vm=u^k−1_m +u^⊥_mandwm=Pl

i=1yi,mei,λ^∗_k∈Eλ^∗_k, whereei,λ^∗_k is an eigenfunction associated withλ^∗_k for 1≤i≤l,lis the multiplicity ofλ^∗_k, andwmcan be consider different from 0 for allm∈N. Note thatkwmk ≤ym, whereym=lmax{|yi,m|; 1≤i≤l}. Using arguments similar to those used in (2.9) and (2.10), we conclude that

kvmk²≤C(1 +kumk)

2(2∗ α−1) 2∗

α +Ckvmk. (2.12)

We can assumekumk ≥1 (ifkumk ≤1, the sequence (u_m) is bounded inH_0,rad¹ (B₁)) and, sincekumk ≤ kvmk+y_m, by (2.12), we obtain

kvmk²≤C(kvmk+y_m)

2(2∗ α−1) 2∗

α +Ckvmk. (2.13)

If ym is bounded, from (2.13), we have that (vm) is bounded in H_0,rad¹ (B1) and, consequently, (um) is bounded inH_0,rad¹ (B1). Now let us assumeym→+∞. Using (2.13), we have

kvm

ym

k²≤Ch(kvmk+ym)

(2∗ α−1)

2∗ α

ym

i² + C

ym

kvm

ym

k

≤Ch 1 y¹⁻

(2∗ α−1)

2∗

m α

kv_m ym

k

(2∗ α−1) 2∗

α + 1

y¹⁻

(2∗ α−1)

2∗

m α

i2

+ C ym

kv_m ym

k.

(2.14)

Thus, we obtain

kvm

ym

k²≤Ckvm

ym

k

2(2∗ α−1) 2∗

α +Ckvm

ym

k+C, which implies the sequence{^v_y^m

m}being bounded because^(2∗α₂⁻¹⁾∗ α

<1, and, by (2.14), k^v_y^m

mk →0 asm→0.

Therefore, possibly up to a subsequence,v_m/y_m→0 a.e. inB₁and strongly in L^q(B1,|x|^α), 1≤q <2^∗_α. Notice that

hJ_λ⁰(u_m),wm

y_mi= 1 y²_m

Z

B1

|∇w_m|²dx−λ Z

B1

|x|^µw_m² dx

− Z

B₁

|x|^α|um|^2∗α−1wm

ym

dx=o(1)

(2.15)

and sincewm∈Eλ^∗_k, we have hJ_λ⁰(u_m),w_m

ym

i=− Z

B₁

|x|^α|u_m|^2∗α−1w_m ym

dx=o(1). (2.16)

Thus, we have Z

B1

|x|^α|um

y_m|^2∗α−2um

y_mwmdx= 1 y²m^∗α−1

Z

B1

|x|^α|um|^2∗α−2um

wm

y_m dx→0 (2.17) as n → ∞. Note, since u_m =v_m+w_m, we have that ^u_y^m

m → w₀ in L^q(B₁,|x|^α) for all 1 ≤ q < 2^∗_α and a.e. in B₁ with w₀ ∈ E_λ^∗

k \ {0}. So, by the Dominated Convergence Theorem and using (2.17), it follows that

Z

B1

|x|^α|um

ym

|^2∗α−2um

ym

wm

ym

dx→ Z

B1

|x|^α|w0|^2∗αdx= 0 (2.18) which is a contradiction. Soy_mis bounded and, consequently, (u_m) is also bounded

inH_0,rad¹ (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 (u_m); thus, by Lemma 2.2, we may assume that (eventually passing to a subsequence)

um* u∈H_0,rad¹ (B1),

u_m→u∈L^p(B₁,|x|^α) for anyp∈[1,2^∗_α[, um→u∈L^p(B1,|x|^µ) for anyp∈[1,2^∗_α[, ifµ≥α,

um→u a.e. inB1.

(2.19)

To check thatuis a solution for (1.1), we need the following lemma.

Lemma 2.3. Let (u_m)be a(P S)_c sequence inH_0,rad¹ (B₁), with c < 2 +α

2(N+α)S(N+α)/(2+α)

α ,

and letvm=um−u. Then vm→0strongly in H_0,rad¹ (B1).

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

kuk²−λkuk²_2,|x|µ− Z

B1

|x|^α|u|^2∗αdx= 0. (2.20) By the Br´ezis-Lieb Lemma [8], it follows that

Z

B₁

|x|^α|um|^2∗αdx= Z

B₁

|x|^α|vm|^2∗αdx+ Z

B₁

|x|^α|u|^2∗αdx+o(1). (2.21) On the other hand, sinceH_0,rad¹ (B1) is a Hilbert Space, we obtain

kumk²=kvmk²+kuk²+o(1). (2.22) By (2.2), (2.21), and (2.22), asu_m→uin L²(B₁,|x|^µ), we obtain

c+o(1) =J_λ(u_m)

=Jλ(u) +1

2kvmk²−λ

2kvmk²_2,|x|µ − 1 2^∗_α

Z

B1

|x|^α|vm|^2∗αdx+o(1)

=J_λ(u) +1

2kv_mk²− 1 2^∗_α

Z

B₁

|x|^α|v_m|^2∗αdx+o(1).

(2.23)

SinceJ_λ⁰(u) = 0 andkvmk²_2,|x|µ =o(1), we conclude that hJ_λ⁰(um), vmi=kvmk²−

Z

B₁

|x|^α|vm|^2∗αdx+o(1).

Then

kvmk²= Z

B1

|x|^α|vm|^2∗αdx+o(1). (2.24) Now, by (2.3) and takingu_mas test function, we note that

Z

B1

|x|^α|um|^2∗αdx=kumk²−λkumk²_2,µ+o(1).

So, asu_m→uinL²(B₁,|x|^µ) and using (2.22), we obtain Jλ(um) =1

2(kumk²−λkumk²_2,|µ|)− 1 2^∗_α

Z

B1

|x|^α|um|^2∗αdx

=1

2(kumk²−λkumk²_2,|µ|)− 1

2^∗_α(kumk²−λkumk²_2,µ+o(1))

= 2 +α

2(N+α)(ku_mk²−λku_mk²_2,|x|µ) +o(1)

= 2 +α

2(N+α)(kuk²−λkuk²_2,|x|µ+kvmk²) +o(1).

(2.25)

From (2.20), we conclude that

kuk²−λkuk²_2,|x|µ ≥0. (2.26)

Thus, by (2.25) and (2.26), we have kvmk²≤ 2(N+α)

2 +α Jλ(um) +o(1).

By (2.2), sincec < _2(N^2+α_+α)S(N+α)/(2+α)

α , formsufficiently large we obtain

kvmk²≤c+o(1)< S_α^{(N+α)/(2+α)}. (2.27) From (1.6) and (2.24), we obtain

kvmk²≤S⁻²

∗ α/2

α kvmk^2∗α+o(1), which implies

kvmk²(S^2∗α/2− kvmk^2∗α−2)≤o(1).

This and (2.27) imply thatvm→0 strongly inH_0,rad¹ (B1).

3. Geometric conditions

Here we prove thatJλsatisfies the geometric condition of Theorem 2.1. Firstly, givenλ >0, we defineλ⁺= min{λ^∗_j : λ < λ^∗_j}and set

H1=⊕[ej]_λ^∗

j≥λ⁺

H_0,rad¹ (B₁)

H2= [e1, . . . , ej]_λ^∗

j<λ⁺. (3.1) Lemma 3.1. There exist δ, ρ >0such that, foru∈H1,

J_λ(u)≥δ if kuk=ρ.

Proof. Let us takeu∈H1, by the variational characterization ofλ⁺ we obtain that Jλ(u)≥1

2 1− λ λ⁺

kuk²−Ckuk^2∗α≥δ >0

whenkuk=ρwithρ >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 functionalJλ.

First, letr∈(0,1) and Br={x∈R^N :|x| ≤r}. We takeξr∈C₀^∞(Br,[0,1]), a radial cut-off function such thatξr= 1 inBr/2 and|∇ξr| ≤4/r, and set u^r(x) = ξr(x)u(x). In [3, Proof of Theorem 3.3] were obtained the following estimates of Br´ezis-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 >0 small enough, we have

(a) ku^rk²=Sα^{(N+α)/(2+α)}+O (N−2)/(2+α)

; (b) ku^rk²₂^∗α∗

α,|x|^α =Sα^{(N+α)/(2+α)}+O ^{(N+α)/(2+α)}

; (c)

ku^rk²_2,|x|µ =







K₁(2+µ)/(2+α) if N >4 +µ;

K1(2+µ)/(2+α)|log|+O (2+µ)/(2+α)

if N = 4 +µ;

K₁(N−2)/(2+α) if N <4 +µ;

(d) ku^rk_1,|x|^µ ≤K2^(N−2)/[2(2+α)]; (e) ku^rk²₂^∗α∗⁻¹

α−1,|x|^α ≤K₃(N−2)/[2(2+α)].

Now we shall prove some main technical lemmas. First of all, we define W(, r) ={u∈H_0,rad¹ (B1);u=u⁻+tu^r, u⁻∈H2, t∈R}.

Remark 4.2. Since u is solution for (1.8), u^r 6∈ [e₁, e₂, . . . , e_k] for any k ∈ N. Thus,W(, r)6=H₂.

Lemma 4.3. If u∈W(, r), then for >0 sufficiently small kuk²₂^∗α∗

α,|x|^α≥ ktu^rk²₂^∗α∗

α,|x|^α−Ct^2∗α(N−2)(N+α)/[(N+2α+2)(2+α)] (4.1) for any t∈R.

Proof. Note that from kuk²₂^∗α∗

α,|x|^α= 2^∗_α Z

B1

|x|^αdx Z u

0

|s|^2∗α−2sds, (4.2) and the Mean Value Theorem, we obtain

kuk²₂^∗α∗

α,|x|^α− ktu^rk²₂^∗α∗

α,|x|^α− ku⁻k²₂^∗α∗ α,|x|^α

= 2^∗_α Z 1

0

ds Z

B1

|x|^α[|tu^r+su⁻|^2∗α−2(tu^r+su⁻)− |su⁻|^2∗α−2su⁻]u⁻dx

= 2^∗_α(2^∗_α−1) Z 1

0

ds Z

B₁

|x|^α|tu^r+τ su⁻|^2∗α−2tu^r·u⁻dx

(4.3)

whereτ =τ(x) is a measurable function such that 0< τ(x)<1.

Using (4.3) and sinceu⁻∈H2, which is a finite-dimension subspace, we obtain kuk²₂^∗α∗

α,|x|^α− ktu^rk²₂^∗α∗

α,|x|^α− ku⁻k²₂^∗α∗ α,|x|^α

≤C Z 1

0

ds Z

B₁

|x|^α(|tu^r|^2∗α−1|u⁻|+|u⁻|^2∗α−1|tu^r|) dx

≤Cktu^rk²₂^∗α∗⁻¹

α−1,|x|^αku⁻k_∞+ku⁻k²_∞,|x|^∗α−1αktu^rk1

≤Cktu^rk²₂^∗α∗⁻¹

α−1,|x|^αku⁻k2+ku⁻k²₂^∗α∗⁻¹

α,|x|^αktu^rk1,

(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

kuk²₂^∗α∗

α,|x|^α− ktu^rk²₂^∗α∗

α,|x|^α− ku⁻k²₂^∗α∗ α,|x|^α

≤Ct^2∗α−1(N−2)/(2(2+α))ku⁻k₂+N+ 2 + 2α 2(N+α) ku⁻k²₂^∗α∗

α,|x|^α+Ct^2∗α(N+α)/(2+α). Finally, again by the Young inequality, we have

kuk²₂^∗α∗

α,|x|^α− ktu^rk²₂^∗α∗

α,|x|^α− ku⁻k²₂^∗α∗ α,|x|^α

≤Ct^2∗α−1

(N−2)

(2(2+α))ku⁻k2^∗_α,|x|^α+N+ 2 + 2α 2(N+α) ku⁻k²₂^∗α∗

α,|x|^α+Ct^2∗α

(N+α) (2+α)

≤Ct^2∗α

(N−2)(N+α) [(N+2α+2)(2+a)] + 1

2^∗_αku⁻k²₂^∗α∗

α,|x|^α+N+ 2 + 2α 2(N+α) ku⁻k²₂^∗α∗

α,|x|^α+Ct^{2∗α(N+α)(2+α)}

=Ct^2∗α

(N−2)(N+α)

[(N+2α+2)(2+α)] +ku⁻k²₂^∗α∗

α,|x|^α+Ct^{2∗α(N+α)(2+α)}

≤Ct^2∗α

(N−2)(N+α)

[(N+2α+2)(2+α)] +ku⁻k²₂^∗α∗ α,|x|^α.

for >0 small enough. The proof is complete.

Lemma 4.4. For >0 sufficiently small, we have ku^rk²−λku^rk²_2,|x|µ

ku^rk²₂∗ α,|x|^α

=







S_α−C(2+µ)/(2+α) if N >4 +µ;

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

S_α+^{(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 (P S)c condition forc below an appropriate constant whenλ=λj for some j∈N, we need to haveN >4 +µ. When N <4 +µ, it is crucial to assume in addition thatλis sufficiently large to prove the (P S)c condition.

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

Lemma 4.5. Assume (1.9), for sufficiently small and positive. If λ6= λ^∗_j, for every j∈N, then

sup

W(,r)

J_λ(u)< (2 +α)

2(N+α)S(N+α)/(2+α)

α . (4.6)

Proof. Note that for fixedu∈H_0,rad¹ (B1) withu6= 0, we obtain sup

t

Jλ(tu) = (2 +α) 2(N+α)

kuk²−λkuk²_2,|x|µ

kuk²₂∗ α,|x|^α

(N+α)/(2+α)

. (4.7)

Since

sup{Jλ(u) :u∈W()\ {0}}

= supn

J_λ(kuk2^∗_α,|x|^α

u kuk2^∗_α,|x|^α

) :u∈W(, r)\ {0}o

≤sup{Jλ(tu) :u∈W(, r)\ {0}with kuk2^∗_α,|x|α = 1 andt∈R}, to show that (4.6) is true, we need to estimate

sup

u∈W(,r),kuk₂∗ α ,|x|α=1

kuk²−λkuk²_2,|x|µ . (4.8) Letu=u⁻+tu^r ∈W(, r) with kuk2^∗_α,|x|^α = 1. By (4.1) and item (b) of Lemma 4.1, forsmall enough, we have

1 =kuk²₂^∗α∗ α,|x|^α

≥ ktu^rk²₂^∗α∗

α,|x|^α−Ct^2∗α(N−2)(N+α)/(N+2α+2)(2+α)

=t^2∗α

S(N+α)/(2+α)

α +O (N−2)/(2+α)

−Ct^2∗α(N−2)(N+α)/(N+2α+2)(2+α)

=t^2∗α

S(N+α)/(2+α)

α +O (N−2)(N+α)/(N+2α+2)(2+α) .

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

kuk²−λkuk²_2,|x|µ

≤ ktu^rk²−λktu^rk²_2,|x|µ+ku⁻k²−λku⁻k²_2,|x|µ

+ 2 Z

B₁

{|tu^r| |∆u⁻|+λ|x|^µ|u⁻||tu^r|}dx

≤ ktu^rk²−λktu^rk²_2,|x|µ+ku⁻k²−λku⁻k²_2,|x|µ+C{ktu^rk1k∆u⁻k∞

+λku⁻k_∞ktu^rk_1,|x|µ}

≤ ktu^rk²−λktu^rk²_2,|x|µ+ku⁻k²−λku⁻k²_2,|x|µ+Cku⁻k2^(N−2)/[2(2+α)]

≤ ktu^rk²−λktu^rk²_2,|x|µ

ktu^rk²₂∗ α,|x|^α

ktu^rk²₂∗

α,|x|^α+ (λ−λ)ku⁻k²_2,|x|µ

+Cku⁻k_2,|x|^µ(N−2)/[2(2+α)],

(4.9)

whereλ= max{λ^∗_j :λ^∗_j < λ}.

Now we defineA(u⁻, , c) = (λ−λ)ku⁻k²_2,|x|µ+Cku⁻k_2,|x|µ^(N−2)/[2(2+α)]. No- tice that

A(u⁻, , c)≤0 or A(u⁻, , c)≤ c²

λ−λ(N−2)/(2+α). (4.10) On the other hand by (4.1) and the boundedness oft, we obtain

ktu^rk²₂∗

α,|x|^α ≤ 1 +C(N−2)(N+α)/[(N+2α+2)(2+α)]2/2^∗_α

≤1 +C^{(N−2)(N+α)/[(N}+2α+2)(2+α)].

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

kuk²−λkuk²_2,|x|µ

≤

Sα−C(2+µ)/(2+α) 1 +C^{[(N−2)(N+α)]/[(N}+2α+2)(2+α)]

+A(u⁻, , c).

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

(N−2)(N+α)

(N+ 2α+ 2)(2 +α) > 2 +µ 2 +α.

Thus,kuk²−λkuk²_2,|x|µ < Sα forpositive and small enough.

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

sup

W(,r)

J_λ(u)< (2 +α)

2(N+α)S(N+α)/(2+α)

α . (4.13)

Proof. WhenN = 4+µ, as for (4.12), from (4.5), (4.9), (4.10) and (4.11), we obtain kuk²−λkuk²_2,|x|µ ≤

Sα−C(2+µ)/(2+α)|log()|+O((2+µ)/(2+α))

×

1 +C[(2+µ+α)(4+µ+α)]/[(6+µ+2α)(2+α)]

+A(u⁻, , c).

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

result.

4.2. Resonant case with N >4 +µ. Now we consider, λ=λ^∗_j for somej∈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 byPj the projector on the eigenspace corresponding toλ^∗_j and set

u˜^r=u^r−P_ju^r. (4.14) Thus, by item (d) in Lemma 4.1, we have

kPju^rk²_2,|x|µ =X

k

Z

B1

|x|^µeku^rdx2

≤Cku^rk²_1,|x|µ ≤C(N−2)/(2+α). (4.15) Consequently, asPju^r is in a finite dimensional space, we obtain

kPju^rk_∞,|x|^µ ≤C(N−2)/2[(2+α)]. (4.16) Furthermore,

ku˜^rk²₂^∗α∗

α,|x|^α− ku^rk²₂^∗α∗ α,|x|^α