In this work we study a class of the critical Kirchhoff-type prob- lems in a Hyperbolic space

Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 53, pp. 1–12.

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

KIRCHHOFF-TYPE PROBLEMS WITH CRITICAL SOBOLEV EXPONENT IN A HYPERBOLIC SPACE

PAULO CESAR CARRI ˜AO, AUGUSTO C ´ESAR DOS REIS COSTA, OLIMPIO HIROSHI MIYAGAKI, ANDR ´E VICENTE

Abstract. In this work we study a class of the critical Kirchhoff-type prob- lems in a Hyperbolic space. Because of the Kirchhoff term, the nonlinearity u^qbecomes “concave” for 2< q <4, This brings difficulties when proving the boundedness of Palais Smale sequences. We overcome this difficulty by using a scaled functional related with a Pohozaev manifold. In addition, we need to overcome singularities on the unit sphere, so that we use variational methods to obtain our results.

1. Introduction In this article we study the Kirchhoff-type problem

− a+b

Z

B³

|∇B³u|²dV_B3

∆_B3u=λ|u|^q−2u+|u|⁴u in H¹(B³), (1.1) where a, b, λare positive constants, 2< q <4,H¹(B³) is the usual Sobolev space on the disc of the Hyperbolic space B³, and ∆_B3 denotes the Laplace Beltrami operator on B³. Problem (1.1) defined in whole spaceR^N, withN ≥3, and with the non-linearity behaving as a polynomial function of degree 2^∗= _N−2^2N was studied by Brezis and Nirenberg [7]. Posteriorly, several authors have studied this class of problems; see for instance Carri˜ao, Costa, and Miyagaki [8].

In the Euclidean context, equation (1.1) is related to a stationary Kirchhoff equation (see [25])

utt−MZ

Ω

|∇xu|²dx

∆xu=f(x, t), (x, t)∈Ω×(0,∞),

where Ω is a bounded domain ofR^N,M(s) =a+bswitha, b >0, andfis a suitable function, which is an extension of the classical D’Alembert’s wave equation. One characteristic of this model is that it considers the effects of the changes in the length of the strings during the vibrations. The main difficulty appears because the equation does not satisfy a pointwise identity any longer. It is generated by the presence of the term containingM in the equation, and it makes (1.1) a nonlocal problem.

Ma and Rivera [27] were the pioneers to study this problem by employing min- imizing methods. In [1], the mountain pass theorem was used, while in [30] the

2010Mathematics Subject Classification. 58J05, 35R01, 35J60, 35B33.

Key words and phrases. Kirchhoff-type problem; variational methods; hyperbolic space.

c

2021. This work is licensed under a CC BY 4.0 license.

Submitted January 30, 2021. Published June 14, 2021.

1

Yang index and critical groups was used. In [21] the equation was studied using the minimization arguments and the Fountain theorem. Results can be seen in [12, 17, 36]. Results involving the Kirchhoff equation and critical exponents can be found in [2, 16, 19, 20, 26] and references therein. See also [11, 13, 14, 32] for some related results.

We also would like to cite the recent works by Xiang, Zhang and Rˇadulescu [38, 39]. In the first one, the authors studied the multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractionalp-Laplacian. In the second paper, they proved the existence of local solution and a blow-up result for a class of nonlocal Kirchhoff diffusion problems.

Our main result reads as follows.

Theorem 1.1. Under the assumptions that2< q <4, forλ >0 sufficiently large, problem (1.1)has a nontrivial solution u∈H¹(B³).

This result extends the result in [20] with respect to the existence in a hyperbolic space. Also, in [9], when a= 1 andb = 0. It also extends [8], where the authors studied (1.1) with 4 < q < 6 for λ > 0 arbitrary. We highlight that the case 2< q <4 is more delicate and it is necessary additional tools.

Finally, we would like to emphasize that an extra difficulty of the present paper is to prove that the Palais Smale sequence is bounded. To overcome this difficulty, we use an appropriated modified functional (seeJ_θ(v) definition in next section). This functional gives us an additional property of the Palais Smale sequence which is fundamental to prove that the sequence is bounded (Lemmas 2.2 and 2.3). Precisely, the scaled functional Jθ works coupled with another appropriated functional, G, which has the propertyG(vk)→0, where (vk) is the Palais Smale sequence. Scaled functional was used by Jeanjean [23] and Jeanjean and Le Coz [24]. See also [19]

and [22].

2. Proof of the main result

For the hyperbolic space Hⁿ, we use the stereographic projection, where each pointP⁰ ∈ Hⁿ is projected to P ∈Rⁿ, whereP is the intersection of the straight line connecting P⁰ and the point (0, . . . ,0,−1). Explicitly the projection operator G:Rⁿ→ Hⁿ andG⁻¹:Hⁿ →Rⁿ given by

G(x) = (x·p(x),(1 +|x|²)p/2) andquadG⁻¹(y) = 1

y_n+1y, x, y∈Rⁿ, wherep(x) = _1−|x|² 2.

We consider the ballB1(0), andBⁿ endowed with the metric ds=p(x)|dx|, where p(x) = 2

1− |x|².

With this notation, the gradient, the Dirichlet integral and the Laplace-Beltrami operator corresponding to this metric are

∇Bⁿu=∇u

p² , Du= Z

D⁰

|∇Bⁿu|²dVBⁿ= Z

D

|∇u|²pⁿ⁻²dx,

∆Bⁿu=p⁻ⁿdiv(pⁿ⁻²∇u).

We denote byD⊂B₁(0) the stereographic projection ofD⁰⊂ Hⁿ. Details involving the hyperbolic space can be found in [3, 18, 31, 33, 34].

Definingv:=p^1/2u, we have thatuis solution of (1.1) if, and only if,v satisfies (a+bkvk²)(−∆v+ (3/4)p²v) =λp^α|v|^q−2v+|v|⁴v, inB1(0)

v= 0, on∂B₁(0), (2.1)

whereα= (6−q)/2 andkvk²=R

B₁(0) |∇v|²+ (3/4)p²v² .

We denote byH_0,r¹ (Ω), Ω :=B₁(0) the subspace ofH₀¹(Ω) of the radial functions which is endowed with the norm

kvk²= Z

Ω

|∇v|²+ (3/4)p²v² .

Since the Euclidean sphere with center at the origin 0∈R^N is also a hyperbolic sphere with center at the origin 0∈Bⁿ,H_0,r¹ (Ω) can also be seen as the subspace of H₀¹(Ω) consisting of the hyperbolic radial functions. See this characterization as well as others remarks in [3, Appendix], for instance, H_0,r¹ (Ω) is embedded compactly in L^q(Ω) for 2< q <2^∗, [3, Theorem 3.1]. Here, we use also [9, Lemma 3.1] and recall that 2^∗= 6.

We consider the functionalJ :H_0,r¹ (Ω)→Rassociated with problem (2.1), J(v) =a

2kvk²+b

4kvk⁴−λ q Z

Ω

p^α|v|^q−1 6

Z

Ω

|v|⁶, (2.2) whose Gateaux derivative is

J⁰(v)w= (a+bkvk²) Z

Ω

∇v· ∇w+3 4p²vw

−λ Z

Ω

p^α|v|^q−2vw− Z

Ω

|v|⁴vw. (2.3) The proof uses variational methods, more exactly, the mountain pass theorem.

To this end, we have the following mountain pass geometry result.

Lemma 2.1 (Mountain pass geometry).

(a) There exist β >0 andρ >0 such that J(v)≥β when kvk=ρ.

(b) There exists an element e∈H_0,r¹ (Ω) with kek> ρsuch thatJ(e)<0.

Proof. (a) We observe that by [9, Lemma 2.1] (see also to [5, 6]) there exists a constantC >0, such that

Z

Ω

p^αv^q ≤CZ

Ω

|∇v|^2q/2

≤ChZ

Ω

|∇v|²+ (3/4)p²v²i^q/2 .

Therefore, J(u)≥a

2kvk²+b

4kvk⁴−Cλ q

hZ

Ω

|∇v|²+ (3/4)p²v²i^q/2

−1 6

Z

Ω

|v|⁶,

and by the Sobolev continuous embedding, there exists a constantC >e 0, satisfying J(u)≥ a

2kvk²+ b

4kvk⁴−Cλ

q kvk^q−Ce

6kvk⁶≥β, where the conclusion follows by makingkvk=ρsufficiently small.

Now, we prove the item (b). We take 0< v∈H_0,r¹ (Ω) and 0< t. Therefore, J(tv) = at²

2 kvk²+bt⁴

4 kvk⁴−λt^q q

Z

Ω

p^α|v|^q−t⁶ 6

Z

Ω

|v|⁶.

ThereforeJ(tv)→ −∞, ast→+∞. Consequently,J satisfies the Mountain Pass

Theorem geometry.

We recall that the pass mountain level is defined by c= inf

γ∈Γ sup

t∈[0,1]

J(γ(t)),

where Γ = {γ ∈ C([0,1], H_0,r¹ (Ω)) : γ(0) = 0, J(γ(1)) <0}. For each θ > 0, we define the functional

J_θ(v) =a 2 Z

Ω

|∇v|²+3 4

1 e^5θp² x

e^2θ

v² +b

4 hZ

Ω

|∇v|²+3 4

1 e^5θp² x

e^2θ

v²i2

−λ q Z

Ω

p^α x e^2θ

v^q−1 6

Z

Ω

|v|⁶.

We also define Φ : R×H_0,r¹ (Ω) → H_0,r¹ (Ω) by Φ(θ, v) = e^θv _e^x2θ

and I : R× H_0,r¹ (Ω)→RbyI(θ, v) =J_θ(Φ(θ, v)).

Using Lemma 2.1, we have that the functional I satisfies the geometry of the Mountain Pass Theorem. Taking

˜ c= inf

˜γ∈˜Γ

sup

t∈[0,1]

I(˜γ(t)), where ˜Γ =

˜

γ∈C([0,1],R×H_0,r¹ (Ω)); ˜γ(0) = (0,0), I(˜γ(1))<0 , we have c = ˜c because Γ ={Φ◦˜γ; ˜γ∈Γ}.˜

Now, we defineG:H_0,r¹ (Ω)→Rby G(v) = 2a

Z

Ω

|∇v|²+9a 8

Z

Ω

p²v²+ 2bZ

Ω

|∇v|²2

+21b 8

Z

Ω

|∇v|² Z

Ω

p²v² +27b

32 Z

Ω

p²v²2

−λ q(q+ 6)

Z

Ω

p^αv^q−2 Z

Ω

|v|⁶.

As it was mentioned in the introduction, the functional Gworks coupled with the scaled functionalJ_θ. The functionalG is a class of Pohozaev functional and it is defined to prove the boundedness of the Palais Smale sequence. The lemma below gives us the main property ofG.

Lemma 2.2. There exists a sequence (vk)⊂H_0,r¹ (Ω) such that J(vk)→c J⁰(vk)→0, G(vk)→0.

Proof. Applying [37, Theorem 2.8] as in [19] and [22, Proposition 4.2], we obtain a sequence (θk, vk) such that

I(θk, vk)→c, I⁰(θk, vk)→0, θk →0.

We note that I(θ, v) = a

2 Z

Ω

∇ e^θv x

e^2θ

2+3 4

1 e^5θp² x

e^2θv² x e^2θ

e^2θ

+ b 4

Z

Ω

∇ e^θv x

e^2θ

2+3 4

1 e^5θp² x

e^2θv² x e^2θ

e^2θ2

−λ q Z

Ω

p^α x e^2θ

e^θv x

e^{2θ q}

−1 6 Z

Ω

e^θv x e^2θ

6

= a 2

e^4θ Z

Ω

|∇v|²+3 4e^3θ

Z

Ω

p²v² + b

4 h

e^8θZ

Ω

|∇v|²² +3

4e^7θ Z

Ω

|∇v|² Z

Ω

p²v²

+ 9 16e^6θZ

Ω

p²v²2i

−λ qe^θ(q+6)

Z

Ω

p^αv^q−1 62^12θ

Z

Ω

|v|⁶. Thus

∂I

∂θ = 2ae^4θ Z

Ω

|∇v|²+9ae^3θ 8

Z

Ω

p²v²+21 b be^7θ

Z

Ω

|∇v|² Z

Ω

p²v² +3

2 9 16e^6θZ

Ω

p²v²2

−λ

q(q+ 6)e^θ(q+6) Z

Ω

p^αv^q−2e^12θ Z

Ω

|v|⁶.

(2.4)

Consideringθk →0, by (2.4) and the definition ofGfor all >0 there existsk0∈N such thatk≥k0

∂I

∂θ(θk, vk)−G(vk)

< . (2.5)

SinceI⁰(θ_k, v_k)→0, by (2.5) we conclude that G(v_k)→0.

On the other hand, sinceI(θ_k, v_k)→candI⁰(θ_k, v_k)→0 we obtain respectively

|I(θk, vk)−J(vk)|< , (2.6)

|I⁰(θk, vk)(ξ, w)−J⁰(vk)(w)|< , (2.7) for all k≥k0. Using the facts thatI(θk, vk)→c andI⁰(θk, vk)→0 by (2.6) and (2.7) we haveJ(vk)→c andJ⁰(vk)→0 respectively.

Next Lemma gives us the boundness for Palais Smale sequence.

Lemma 2.3. The sequence(vk)⊂H_0,r¹ (Ω) obtained in Lemma2.2 is bounded.

Proof. We note that J(vk)−G(vk) =a1

2− 2 q+ 6

Z

Ω

|∇vk|²+3a 8

1− 3

q+ 6 Z

Ω

p²v²_k +b1

4− 2 q+ 6

Z

Ω

|∇vk|²+3b 8

1− 7

q+ 6 Z

Ω

|∇vk|² Z

Ω

p²v²_k + 9b

64

1− 6 q+ 6

Z

Ω

p²v_k²2

+ 2 q+ 6 −1

6 Z

Ω

|vk|⁶.

Since all the coefficients of the terms involving the integrals, on the right side of the equality are positive, J(vk)→c and G(vk)→0 by Lemma 2.2, we have (vk)

bounded.

In next lemma, the numberSis the best constant of Sobolev (see [35]). We follow the arguments of [7]. See also [9, 20, 19, 28]. We are going to omit some calculus, the reader can found the details in [8] where was studied the case 4< q <6.

Lemma 2.4. We have c < ¹₄abS³+₂₄¹b³S⁶+₂₄¹(b²S⁴+ 4aS)^3/2, where S:= inf

u∈H¹_0,r(Ω)

R

Ω|∇u|² R

Ωu^61/3.

Proof. First, we observe that it is sufficient to show that there exists a v0 ∈ H_0,r¹ (Ω), v06= 0, such that

sup

t≥0

J(tv₀)< 1

4abS³+ 1

24b³S⁶+ 1

24(b²S⁴+ 4aS)^3/2. (2.8)

Indeed, observing that J(tv0) → −∞ as t → ∞, there exists R > 0 such that J(Rv0)<0. Now, we writeu1:=Rv0, and from Lemma 2.1, we have

0< β≤c= inf

γ∈Γ max

τ∈[0,1]J(γ(τ))≤sup

t≥0

J(tv0)<1

4abS³+1

24b³S⁶+1

24(b²S⁴+4aS)^3/2. Therefore, we are going to prove the existence of a functionv0such that (2.8) holds.

We consider 0< R < ¹₂ a fixed number and letϕ∈C₀^∞(Ω) be a cut-off function with support at B2R, such that ϕis identically 1 onBR and 0 ≤ϕ≤1 on B2R. Here,Brdenotes the ball in R³ with center at the origin and radiusr.

Givenε >0 we setψε(x) :=ϕ(x)ωε(x), where ωε(x) = (3ε)^1/4 1

(ε+|x|²)^1/2, andω_ε satisfies (see[35])

Z

R³

|∇ω_ε|²= Z

R³

|ω_ε|⁶=S^3/2. (2.9)

From the definition ofωε, it can be shown that Z

BR

|∇ωε|²≤ Z

BR

|ωε|⁶, (2.10)

Z

B₁−BR

|∇ψε|²=O(ε^1/2) as ε→0. (2.11) Now, we define

vε:= ψε

R

B_2Rψ_ε^61/6 andXε:=R

B₁|∇vε|². Then, we have Xε=

Z

B_R

|∇ψε| B² +

Z

B_2R−BR

|∇ψε| B² , whereB:= R

B2Rψ_ε⁶1/6

. Thus, sinceϕ≡1, and consequently∇ϕ≡0 onB_R, we have

Xε= 1 B²

Z

BR

|∇ωε|²+ Z

B2R−BR

|∇ψε|². By (2.10) and (2.11) we obtain

Xε≤S+O(ε^1/2). (2.12)

On the other hand, we have

t→+∞lim J(tv_ε) =−∞, ∀ε >0.

This implies that there existstε>0 such that sup

t≥0

J(tv_ε) =J(t_εv_ε). (2.13) Now, we are going to prove an estimate fort_ε. From (2.13), we have

d

dtJ(tvε)_|t=tε = 0, thus,

atεkvεk²+bt³_εkvεk⁴−λt^q−1_ε Z

Ω

p^α|vε|^q−t⁵_ε Z

Ω

|vε|⁶= 0,

which implies

akvεk²+bt²_εkvεk⁴−λt^q−2_ε Z

Ω

p^α|vε|^q−t⁴_ε Z

Ω

|vε|⁶= 0.

SinceR

Ω|vε|⁶= 1, we have

−akvεk²−bt²_εkvεk⁴+t⁴_ε≤0.

Hence

0≤t²_ε≤bkv_εk⁴+

(bkv_εk⁴)²+ 4akv_εk^21/2

2 :=t₀.

Since the functiont 7→ ^a₂t²kvεk²+₄^bt⁴kvεk⁴−^t₆⁶ is increasing on [0, t0), denoting C₁=akvεk²and C₂=bkvεk⁴, we have

J(tεvε)≤ C₁C₂ 4 +C₂³

24 + 1

24(C₂²+ 4C1)^3/2−λt^q_ε q

Z

Ω

p^αv^q_ε. Considering A= 3/4R

Ωp²v²_ε, by definition of the norm, and the inequality (2.12), we obtain

J(tεvε)≤ab

4(Xε+A)³+ b³

24(Xε+A)⁶+ 1 24

b²(Xε+ 4)⁴+ 4a(Xε+A)3/2

−λt^q_ε q

Z

Ω

p^αv^q_ε

≤ab

4(S+O(ε^1/2) +A)³+ b³

24(S+O(ε^1/2) +A)⁶ + 1

24 h

b²(S+O(ε^1/2) +A)⁴+ 4a(S+O(ε^1/2) +A)i^3/2

−λt^q_ε q

Z

Ω

p^αv^q_ε.

Using several times the standard inequality (see e.g. [28, Page 778]) (a+b)^β≤a^β+β(a+b)^β−1b, ∀β ≥1, ∀a, b >0, we infer that

J(tεvε)≤ abS³

4 +b³S⁶ 24 + 1

24(b²S⁴+ 4aS)^3/2+O(ε^1/2) +

Z

B2R

3C

4 p²v²_ε−λCεp^αv_ε^q ,

(2.14)

for some constantC >0, whereC_ε=t^q_ε/q.

At this point, we can assume that there exists a positive constantC₀ such that C_ε≥C₀>0 for all ε >0. If it is not true, then we can find a sequenceε_k →0 as k → ∞, such that t_εk →0 as k→ ∞, sinceC_ε ≥0. Now, up to a subsequence, that we still denote byε_k, we havet_εkv_εk→0, ask→ ∞. Therefore,

0< c≤sup

t≥0

J(tvε_k) =J(tε_kvε_k) =J(0) = 0, which is a contradiction.

Observing thatR

B_2Rp²v²_ε<∞, we claim that

ε→0lim 1 ε^1/2

Z

B_2R

3C

4 p²v²_ε−C_ελp^αv^q_ε

=−∞.

Assuming the Claim is proved, from (2.14) we have J(tεvε)< abS³

4 +b³S⁶ 24 + 1

24(b²S⁴+ 4aS)^3/2, for someε >0 sufficiently small, and the proof is complete.

Now, we prove the Claim. For this, it is sufficient to show that

ε→0lim 1 ε^1/2

Z

B_R

3C

4 p²ω_ε²−C_ελp^αω_ε^q

=−∞ (2.15)

Z

B2R−BR

3C

4 p²v_ε²−Cελp^αv^q_ε

=O(ε^1/2). (2.16) First, we consider

Jε= 1 ε^1/2

Z

BR

3C

4 p²ω_ε²−Cελp^αω_ε^q

= 3C 4ε^1/2

Z

B_R

2 1− |x|²

² (3ε)^1/2

(ε+|x|²)−λC_ε ε^1/2 Z

B_R

2 1− |x|²

^α (3ε)^q/4 (ε+|x|²)^q/2

= ˜C Z

BR

2 1− |x|²

2 1

(ε+|x|²)−λC˜εε^(q−2)4 Z

BR

2 1− |x|²

α 1 (ε+|x|²)^q/2

=J₁−J₂,

(2.17) for some constantC >e 0. We observe that on BR,

2< 2

1− |x|² ≤ 2

1−R². (2.18)

Therefore, making the change of variables x= ε^1/2y and using the polar coordi- nates, we obtain

J₁≤ 4 ˜C

(1−R²)²ωε^1/2

Z Rε^−1/2

0

r²

(1 +r²)dr, (2.19) for some constantC >e 0. Similarly, forJ₂, we have

J₂≥λC˜_ε2^αwε^−q4+1

Z Rε^−1/2

0

r²

(1 +r²)^q/2dr, (2.20) whereCe_εis a positive constant. Thus, combining (2.17), (2.19) and (2.20) we obtain

J_ε≤ 4 ˜C

(1−R²)²ωε^1/2

Z Rε^−1/2

0

r² (1 +r²)dr

−λC˜ε2^αwε^−q4+1

Z Rε^−1/2

0

r² (1 +r²)^q/2dr.

(2.21)

Observing that

Z Rε^−1/2

0

r²

1 +r²dr=Rε^−1/2−tan⁻¹(Rε^−1/2) we obtain

J_ε≤C−Cε^1/2tan⁻¹(Rε^−1/2)−λCε^−q4+1

Z Rε^−1/2

0

r²

(1 +r²)^q/2dr. (2.22)

Now, as

Z Rε^−1/2

0

r²

(1 +r²)^q/2dr≥

Z Rε^−1/2

0

1

1 +r²dr≥C >0,

for allε < ε0, withε0 small enough. At this moment, it is possible to see the main difference with the proof of [8, Lemma 2.3]. To control the sign of the expression of (2.15) it is necessary to use the assumption involvingλ. Since, by assumption,λ is positive and sufficiently large, we can takeλ=ε⁻¹² and we conclude that (2.15) holds.

The proof of (2.16) is the same of [8, (2.13)], This completes the proof.

3. Proof of Theorem 1.1

Let{vn}be the sequence given by Lemma 2.2. Lemma 2.3 implies that {vn} is bounded inH_0,r¹ (Ω). Thus, we can assume, passing to a subsequence, thatvn* v, weakly inH_0,r¹ (Ω) asn→ ∞. Arguing as in [9], we have

J⁰(v_n)w=o(1), ∀w∈H_0,r¹ (Ω). (3.1) Now, we observe that

|J⁰(vn)w−J⁰(v)w| →0, (3.2) as n → ∞, for all w ∈ C_c,rad^∞ (Ω). From this, it follows that J⁰(v)w = 0, for all w∈C_c,rad^∞ (Ω). By denseness, we conclude that

J⁰(v)w= 0, ∀w∈H_0,r¹ (Ω), (3.3) andv is a critical point of the functionalJ restricted to the spaceH_0,r¹ (Ω).

Now, we follow the ideas in [4, 10, 15] (see also [29]). Since H_0,r¹ (Ω) is a closed subspace ofH₀¹(Ω), we can write

H₀¹(Ω) =H_0,r¹ (Ω)⊕H_0,r¹ (Ω)^⊥,

where ·^⊥ denotes the orthogonal complement of the space. Therefore, for each w∈H₀¹(Ω), there existϑ∈H_0,r¹ (Ω) andϑ^⊥∈H_0,r¹ (Ω)^⊥ such that

w=ϑ+ϑ^⊥. (3.4)

AsH_0,r¹ (Ω) is a Hilbert space and J⁰(v)∈H_0,r¹ (Ω)^∗, from the Riesz Representa- tion Theorem there existsz∈H_0,r¹ (Ω) such that

J⁰(v)w= Z

Ω

∇z· ∇w, ∀w∈H_0,r¹ (Ω).

Thus, asz∈H_0,r¹ (Ω) andϑ^⊥∈H_0,r¹ (Ω)^⊥, we have

J⁰(v)ϑ^⊥= 0. (3.5)

From (3.3), (3.4) and (3.5), for eachw∈H₀¹(Ω), we obtain J⁰(v)w=J⁰(v)ϑ+I⁰(v)ϑ^⊥= 0.

This allows us to conclude that v is a critical point of the functional J in H₀¹(Ω) and consequentlyvis a weak solution for problem (2.1).

If v 6= 0 we are done. Now, we suppose that v ≡ 0. Considering v_n * 0, as n→ ∞, we have

J⁰(vn)vn=akvnk²+bkvnk⁴−λ Z

Ω

p^α|vn|^q− Z

Ω

|vn|⁶=on(1). (3.6)

By [9, Lemma 3.1], we obtain λ

Z

Ω

p^α|vn|^q →0, asn→ ∞, (3.7) LetL₁>0,L₂>0 be such that

akvnk²→L1 and bkvnk⁴→L2, as n→ ∞. (3.8) By (3.6), (3.7), and (3.8),

Z

Ω

|vn|⁶→L1+L2, asn→ ∞. (3.9) But

SZ

Ω

v_n^61/3

≤ Z

Ω

|∇vn|², (3.10)

which implies aSZ

Ω

v⁶_n1/3

≤a Z

Ω

|∇v_n|²≤a Z

Ω

|∇v_n|²+ (3/4)p²v²_n

=akv_nk², (3.11) bS²Z

Ω

v⁶_n2/3

≤bhZ

Ω

|∇v_n|²i2

≤bhZ

Ω

|∇v_n|²+ (3/4)p²v_n²i2

=bkv_nk⁴. (3.12) Thus, by (3.8), (3.9), (3.11) and (3.12),

L₁≥aS(L₁+L₂)^1/3 and L₂≥bS²(L₁+L₂)^2/3. (3.13) On the other hand,J(vn) =c+o(1). So

c=L₁ 2 +L₂

4 −1

6(L₁+L₂) =L₁ 3 +L₂

12. (3.14)

By (3.13) we have

(L1+L2)^1/3≥ bs²+ (b²s⁴+ 4as)^1/2

2 . (3.15)

Hence by (3.13), (3.14) and (3.15), c≥ 1

3L₁+ 1

12L₂≥1

3aS(L₁+L₂)^1/3+ 1

12bS²[(L₁+L₂)^1/3]²

≥ 1

4abS³+ 1

24b³S⁶+ 1

24(b²S⁴+ 4aS)^3/2,

which is a contradiction to Lemma 2.4. Therefore, we conclude thatv6= 0.

Acknowledgments. This work was done while P. C. C. was visiting the Depart- ment of Mathematics of UFJF, under financial support by FAPEMIG CEX APQ 00063 15. A. C. R. C. was supported in part by PNPD CAPES 2017 PGM/UFJF.

O. H. M. received research grants from CNPq/Brazil 307061/2018-3, FAPEMIG CEX APQ 00063/15 and INCTMAT/CNPQ/Brazil. The authors would like to thank the anonymous referees for all insightful comments, which allow us to im- prove our original version.

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Paulo Cesar Carri˜ao

Departamento de Matem´atica, Universidade Federal de Minas Gerais, Belo Horizonte, MG 31270-901, Brazil

Email address:[email protected]

Augusto C´esar dos Reis Costa

Faculdade de Matem´atica, Instituto de Ciˆencias Exatas e Naturais, Universidade Fed- eral do Par´a, Bel´em, PA 66075-110, Brazil

Email address:[email protected]

Olimpio Hiroshi Miyagaki

Departamento de Matem´atica, Universidade Federal de Juiz de Fora, Juiz de Fora, MG 36036-330, Brazil

Email address:[email protected]

Andr´e Vicente

Centro de Ciˆencias Exatas e Tecnol´ogicas, Universidade Estadual do Oeste do Paran´a, Cascavel, PR 85819-110, Brazil

Email address:[email protected]