On critical Kirchhoff problems driven by the fractional Laplacian

https://doi.org/10.1007/s00526-021-02065-8

Calculus of Variations

On critical Kirchhoff problems driven by the fractional Laplacian

Luigi Appolloni¹·Giovanni Molica Bisci²·Simone Secchi¹

Received: 23 April 2021 / Accepted: 16 July 2021 / Published online: 27 August 2021

© The Author(s) 2021

Abstract

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the Sobolev embedding sense. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.

Mathematics Subject Classification 35S15·35J20

1 Introduction

The equation that goes under the name ofKirchhoff equationwas proposed in [17] as a model for the transverse oscillation of a stretched string in the form

ρh∂_tt²u−

p0+ ^Eh 2L

_L

0

|∂xu|² d x

∂_{x x}² u+δ ∂tu+ f(x,u)=0 (1) fort ≥0 and 0< x < L, whereu = u(t,x)is the lateral displacement at timetand at positionx,E is the Young modulus,ρis the mass density,his the cross section area,Lthe length of the string,p₀is the initial stress tension,δthe resistance modulus andgthe external force. Kirchhoff actually considered only the particular case of (1) withδ= f =0.

Communicated by P. H. Rabinowitz.

The first and third author are supported by GNAMPA, project “Equazioni alle derivate parziali: problemi e modelli”.

B

l.appolloni1@campus.unimib.it Simone Secchi

simone.secchi@unimib.it

1 Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Milan, Italy 2 Dipartimento di Scienze Pure e Applicate, Università di Urbino Carlo Bo, Urbino, Italy

Through the years, this model was generalized in several ways that can be collected in the form

∂_tt²u−M(u²)Δu= f(t,x,u), x∈Ω (2) for a suitable functionM: [0,∞)→R, calledKirchhoff function. The setΩis a bounded domain ofR^N, andu² = ∇u²₂denotes the Dirichlet norm ofu. The basic case corre- sponds to the choice

M(t)=a+bt^γ−1, a≥0, b≥0, γ ≥1.

WhenM(0) =0, i.e.a =0, the equation is calleddegenerate.Stationary solutionsto (3) solve the equation

−M(u²)Δu= f(x,u), x∈Ω

u=0 on∂Ω (3)

We refer to [26] for a recent survey of the results connected to this model.

The existence and multiplicity of solutions to Kirchhoff problems under the effect of a critical nonlinearity f have received considerable attention. The term critical refers here to the rough assumption that f(u)∼ |u|^2∗−2uwith 2^∗=2N/(N −2). The natural setting of the corresponding equation inH₀¹(Ω)yields alack of compactness, since the embedding of H₀¹(Ω)intoL^2∗(Ω)is only continuous. Straightforward techniques of Calculus of Variations fail, and more advanced results from Critical Point Theory must be used. In particular, P.- L. Lions’ Concentration-Compactness appears as a natural tool for the analysis of the loss of compactness.

The relevant outcome is that the Kirchhoff functionMinteracts with the critical growth of the nonlinearityg: the validity of the Palais–Smale compactness condition holds only under a condition like

a^N−42 b≥C₂(N),

and a similar inequality ensures that the associated Euler functional is weakly lower semi- continuous. For some very recent results on Kirchhoff-type problems, see [15,18] as well as [27] for related topics.

In a very recent paper, Faraci and Silva (see [13]) obtained several quantitative results for the problem

⎧⎨

⎩

−

a+b

Ω|∇u|²d x

Δu= |u|^2∗−2u+λg(x,u) inΩ

u=0 on∂Ω

(4)

whereΩ is an open bounded subset ofR^N,N > 4,aandba repositive fixed numbers, λis a parameter andgis a Carathéodory function that satisfies suitable growth conditions.

By using a fibering-type approach, the authors of [13] investigate existence, non-existence and multiplicity of solutions to (4). In a previous paper, see [14], Faraci, Farkas and Kristály studied Eq. (4) withg(x,u)= 0 and under suitable assumptions on the parametersaand bthey proved that the functional associated to the problem is sequentially weakly lower semicontinuous, satisfies the Palais–Smale condition and is convex.

The purpose of the present paper is to extend part of these results to thefractionalcoun- terpart of the Kirchhoff problem

⎧⎨

⎩

a+b

Q

|u(x)−u(y)|²

|x−y|^n+2s d x d y

(−Δ)^su= |u|^2∗s−2u+λg(x,u) inΩ

u=0 inR^N\Ω

(P_a,b^λ ) where Ω ⊂ R^N is a bounded domain with Lipschitz boundary∂Ω,Q = R^2N \O andO = Ω^c×Ω^c,a andbare strictly positive real numbers,s ∈ (0,1), N > 4sand 2^∗_s :=2N/(N −2s)denotes the critical exponent for the Sobolev embedding ofH^s(R^N) into Lebesgue spaces.gis a function that satisfies hypothesis similar to the one in (4) adapted to the non local case. The fractional Laplacian in (P_a,b^λ ) is defined as

(−Δ)^su(x)=K_N,s lim

→0⁺

R^N\B(0)

u(x)−u(y)

|x−y|^N+2s d y where

1 K_N,s :=

R^N

1−cosζ1

|ζ|^N+2s dζ.

Since the parametersis given, we will work with a rescaled version of the operator and this enables us to assume thatK_N,s =1. For references about the fractional Laplacian we refer to [11], [1] and to the monograph [23]. We define the spaceX as the set of functions u:R^N →Rsuch thatu|Ω∈L²(Ω)and

(x,y)→ u(x)−u(y)

|x−y|^N/2+s

∈L²(Q), endowed with the norm

uX= u_L²_(Ω)+

Q

|u(x)−u(y)|²

|x−y|^N+2s d x d y ¹₂

. (5)

We also set

X₀^s(Ω):=

u∈X: u=0 a.e. inR^N \Ω . We introduce the best Sobolev constant as

S_N,s := inf

u∈X₀^s(Ω)

u² u²₂∗ s

(6) where

u²:=

Q

|u(x)−u(y)|²

|x−y|^N+2s d x d y.

The norm introduced in the previous equation is induced by the scalar product u, vX^s₀(Ω):=

Q

(u(x)−u(y))(v(x)−v(y))

|x−y|^N+2s d x d y for allu, v∈X^s₀(Ω) and we recall that inX₀^s(Ω)it is equivalent to (5). For further details we refer the reader to [28, Lemma 6]. As it is easy to check, looking for solution of (P_a,b^λ ) is equivalent to finding the critical points of the functionalI_a,b^λ : X^s₀(Ω)→Rassociated to the problem:

I_a,b^λ (u):=a

2u²+b

4u⁴− 1 2^∗_su²₂^∗s∗

s −λ

ΩG(x,u)d x where we denote withG(x,t)=_t

0 g(x, ω)dω. Arguing as in [30, Proposition 1.12], we get I_a,b^λ

(u)[v]=

a+bu²

u, vX^s₀(Ω)−

Ω|u|^2∗s−2uvd x−λ

Ωg(x,u)vd x (7) for allu, v∈X^s₀(Ω). When we haveg(x,u)=0 we will use the notation

Ia,b(u):= a

2u²+b

4u⁴− 1 2^∗_su²₂^∗s∗

s

and we point out thatI_a,bis aC²-functional.

The interest in generalizing to the fractional case the model introduced by Kirchhoff does not arise only for mathematical purposes. In fact, following the ideas of [6] and the concept of fractional perimeter, Fiscella and Valdinoci proposed in [16] an equation describing the behaviour of a string constrained at the extrema in which appears the fractional length of the rope. The interested reader can also consult [7–9] and the references therein for further motivations and applications of operators similar to the one proposed in (P_a^λ_,b).

Recently, problem similar to (5) has been extensively investigated by many authors using different techniques and producing several relevant results. In [16] Fiscella and Valdinoci showed the existence of a non-negative solution of mountain pass type for an equation with a critical term perturbed with a subcritical nonlinearity. With the same spirit of the previous one, in their seminal paper [4], Autuori, Fiscella and Pucci generalize these results to the degenerate case, i.eM(0)=0, without requiring monotonicity assumption on the function M. We stress that in these two articles the operator taken into account is more general to the one we consider here, but the two coincide making a particular choice on the kernel; see also the paper [24] due to Molica Bisci and Vilasi. In the recent [21], Liu, Squassina and Zhang studied ground state solutions for the Kirchhoff equation plus a potential with a non linear term asymptotic to a power with critical growth in low dimension. It is also worth mentioning [22] where Mingqi, Rˇadulescu and Zhang proved the existence of nontrivial radial solutions in the non-degenerate and degenerate cases for the non local Kirchhoff problem in which the fractional Laplacian is replaced by the fractional magnetic operator.

Despite all the results cited above, to the best of our knowledge, in literature there are still no articles summarizing the situation of different kind of solutions at different level of energy for the fractional Kirchhoff problem. Furthermore, even if some of the results we are going to prove are known, we present a proof based on a adaptation to the fractional case due to Palatucci and Pisante ([25]) of the Lions second concentration-compactness principle; for the original version of the lemma we refer to [20], as well as [19].

For the reader’s convenience, we collect here our main results.

Theorem 1 Define

L_N,s :=4s(N−4s)^N−4s2s N^N−2s2s S

N 2s N,s

, PSN,s := 2s(N−4s)^N−4s2s (N−2s)^N−2s2s S

N 2s N,s

,

and

C_N,s := 2s(N −4s)^N−4s2s (N +2s)^N−2s2s (N−2s)^N−2ss S

N 2s N,s

.

The following assertions holds true:

(i) the energy functionalIa,bis sequentially weakly lower semicontinuous on X^s₀(Ω)if and only if a^N−4s2s b≥L_N,s.

(ii) If a^(N−4s)/2sb≥PSN,s, the functional_Ia,bsatisfies the compactness Palais–Smale condition at level c∈R.

(iii) If a^(N−4s)/2sb≥CN,s, then the functionalIa,bis convex on X₀^s(Ω).

Theorem1guarantees the validity of some crucial properties such as the sequentially weakly lower semicontinuity and the Palais–Smale condition. As we are going to see in the next statement, these facts enable us to use traditional variational methods to completely describe the situation for problem (P_a,b^λ ). We begin providing two results about the existence of global minimizers at different level of energy.

Theorem 2 Let a,b∈R⁺such that a^(N−4s)/2sb≥LN,sand set ι^s_λ:=inf

I_a,b^λ (u)| u∈X^s₀(Ω)\ {0}

for anyλ >0.

There existsλ^s₀ ≥0such that for anyλ > λ^s₀it is possible to find u^s_λ ∈ X₀^s(Ω)\ {0}such thatI_a,b^λ (u^s_λ)=ι^s_λ<0.

Theorem 3 Letλ=λ^s0. The following statements hold:

(i) if a^(N−2s)/2sb>L_N,sthen there exists u^s_λ∈X^s₀(Ω)\ {0}such thatι^s_λs

0 =I_a,b^λs0 =0;

(ii) if a^(N−2s)/2sb=L_N,s, then u=0in the only minimizer forι^s_λs 0

.

In the next Theorem we give some information on what happens when we do not keep fixed the parametersa, andb. It asserts we have some kind of stability when the product a^(N−4s)/2sbbecomes close toL_N,s.

Theorem 4 Let (ak)k,(bk)k be a sequence of real positive numbers such that ak → a, bk → b and a^(N−4s)/2s_k bk LN,s. Setting λk := λ^s₀(ak,bk) we have thatλk → 0as k→ ∞. Furthermore, if(uk)k⊂X^s₀(Ω)\ {0}such thatλk=λ^s₀(uk)then uk0and

uk²₂∗ s

u_k² →S_N,s.

Next statement shows the existence of solution of mountain pass type whenλ≥λ^s₀. Theorem 5 Ifλ ≥ λ^s₀, then there exists av_λ^s ∈ X₀^sΩ\ {0}such thatI_a,b^λ (v_λ^s) = c^s_λand I_a^λ_,b

(v^s_λ)=0where

c^s_λ:= inf

h∈Γ_λ^s max

ζ∈[0,1]I_a,b^λ (h(ζ )) and

Γ_λ^s :=

h∈C

[0,1],X^s₀(Ω)

|h(0)=0, h(1)=u^s

λ^s0

.

Finally we focus on the caseλ∈(λ^s₀−δ, λ^s₀)for someδ >0 small.

Theorem 6 Set

ˆ

ι^s_λ:=inf{I_a,b^λ (u)|u∈X^s₀(Ω), u ≥r}

for some r >0. There existδ,r>0such that for anyλ∈(λ^s₀−δ, λ^s₀)the valueˆι^s_λis attained at a functionw^s_λ∈X^s₀(Ω)satisfyingw^s_λ>r .

Theorem 7 For anyλ∈(λ^s0−δ, λ^s0)there isv_λ^s ∈X₀^s(Ω)\ {0}such thatI_a^λ_,b(v_λ^s)=c_λ^sand I_a,b^λ

(v^s_λ)=0where

c^s_λ:= inf

h∈Γ_λ^s max

ζ∈[0,1]I_a^λ_,b(h(ζ )) and

Γ_λ^s := {h∈C

[0,1],X₀^s(Ω)

:h(0)=0, h(1)=w^s_λ}

Our paper is organized as follows: in Sect.1we present the classic Kirchhoff model, its generalization to the non local case and we collect in a synthetic way our main results. In Sect.2we prove for the functional associated to the problem withg(x,u)=0 the weak lower semicontinuity, the validity of the Palais–Smale condition and the convexity under suitable assumption on the parametersaandb. Since the perturbationgwill have a subcritical growth, we decided to prove these conditions for the problem with the pure power in order to ease notation. We stress that the functional associated to the perturbed problem still verifies these properties and proofs need only minor adjustments. In Sect.3we prove the existence of global minimizers, local minimizers and mountain pass type solutions with different energy level at varying of the parameterλ. At the end of Sect.3, strengthening the hypothesis on the non linear termg, we are able to give also a non existence result for problem (P_a,b^λ ).

2 Semicontinuity and the validity of the Palais–Smale condition

In this section we completely describe the range of parametersaandbfor which the functional Ia,bassociated to the problem

⎧⎨

⎩

a+b

Q

|u(x)−u(y)|²

|x−y|^n+2s d x d y

(−Δ)^su= |u|^2∗s−2u inΩ

u=0 inR^N \Ω

(P_a,b)

is (sequentially) weakly lower semicontinuous.

Proof of Theorem1 (i)We assume thata^N−4s2s b≥L_N,s, and we choose a sequence(u_n)n⊂ X^s₀(Ω)such thatu_nu. Since the embeddingX₀^s(Ω) →L^p(Ω)is compact (see for instance [29, Lemma 9]),unconverges toustrongly inL^p(Ω)for anyp∈

1,2_s^∗

. We notice that u_n−u²+2u_n−u,uX^s₀(Ω)= u_n,u_nX^s₀(Ω)+ u,uX₀^s(Ω)−2u_n,uX^s₀(Ω)

+2un,uX₀^s(Ω)−2u,uX^s₀(Ω)= un²− u². (8) Hence

un²− u²= un−u²+2un−u,uX^s₀(Ω)= un−u²+o(1)

asn→ ∞. After that, we compute un⁴− u⁴=

un²− u² un²+ u²

=

un−u²+o(1) un−u²+2u²+o(1)

. (9)

Finally, using the Brezis-Lieb Lemma (see [5, Theorem 1]), we have u_n−u²₂^∗s∗

s = u_n²₂^∗s∗

s − u²₂^∗s∗

s +o(1) (10)

asn→ ∞. Putting together (8), (9), (10) and the Sobolev inequality (6) we obtain Ia,b(u_n)−Ia,b(u)=a

2

u_n²− u² +b

4

|u_n⁴− u⁴

− 1 2^∗_s

u_n²₂^∗s∗

s − u²₂^∗s∗ s

=a

2un−u²+b 4

un−u⁴+2u²un−u²

− 1

2^∗_su_n−u²₂^∗s∗ s +o(1)

≥a

2u_n−u²+b

4u_n−u⁴− S⁻

2∗ 2s N,s

2^∗_s u_n−u^2∗s +o(1)

= un−u² a

2+b

4un−u²−S_N,s

2^∗_s un−u^2∗s−2

+o(1) (11) asn→ ∞. At this point, we introduce the auxiliary function

f_N,s(ζ )= a 2 +b

4ζ²− S⁻

2∗ 2s N,s

2^∗_s ζ^2∗s−2, ζ ≥0. It is easy to verify that the function f_N,sattains its minimum at the point

m_N,s= b

2 2^∗_s 2^∗_s−2S

2∗s N,s2

_2∗¹

s−4

, and that

a^N−4s2s b≥LN,s⇔ fN,s(mN,s)= 1 2

a−b^−N−4s2s L

N−4s2s N,s

≥0 (12) From (11) and (12) it follows that

lim inf

n→∞

I_a,b(u_n)−I_a,b(u)

≥lim inf

n→∞ u_n−u²f_N,s(u_n−u)≥0, which concludes this part of the proof.

Conversely, we proceed by contradiction, assuming that the functionalIa,bis sequentially weakly lower semicontinuous but

a^N−4s2s b<LN,s (13)

Let{un}n ⊂ X₀^s(Ω)be a minimizing sequence forSN,s. By homogeneity we may assume furthermore thatun2^∗_s =1 for everyn, so that we deduce that the sequence{un}n must be bounded. Up to a subsequence, we have thatunu inX₀^s(Ω)for someu ∈ X₀^s(Ω)\ {0}. Besides, exploiting the weak lower semicontinuity of the norm, we have that u ≤

lim inf_n→∞u_n =:Land there exists a subsequence{u_nk}ksuch thatL=lim_k→∞u_nk. We point out thatL > 0, sinceu =0. NowN > 4simplies that 0 <2^∗_s −2 < 2, and lim_x→+∞ f_N,s(x)= +∞. As we have already seen, the function f_N,sattains its minimum at the pointmN,sand this, together with (13), impliesf_N,s(mN,s) <0. Setc=m_N,s/L>0.

We notice that lim inf

n→∞ I_a,b(cun)≤lim inf

k→∞ I_a,b(cunk)

=lim inf

k→∞ cunk²fN,s(cunk)=(cL)²f_N,s(cL)

=(cL)²f_N,s(m_N,s)≤ cu²f_N,s(m_N,s)≤ cu²f_N,s(cu). (14) We also have that

cu²fN,s(cu)=a

2cu²+b

4cu⁴− S_N^2∗s_,s^/2 2^∗_s cu

≤a

2cu²+b

4cu⁴− 1 2^∗_s

Ω|cu|^2∗sd x=I_a,b(cu). (15) Comparing (13) with (14) we get

lim inf

n→∞ Ia,b(cu_n)≤Ia,b(cu). (16)

We claim that a strict inequality holds in (16). Indeed, if we had equality, the functioncu would attain the minimum in (6). This is impossible, sinceΩ=R^N(see [10, Theorem 1.1]).

The proof is complete.

Proof of Theorem1 (ii)Let{un}n ⊂ X^s₀(Ω)be a(P S)csequence, i.e.Ia,b(un) →cand I_a,b (u_n)→0 asn→ ∞. Recalling (6), we observe that

Ia,b(u)=au²+bu⁴−

Ω|u|^2s∗d x≥au²+bu⁴−S⁻

2∗s 2 N,s u^2∗s.

Since 2^∗_s <4 we have thatI_a,bis coercive, and from that we can deduce the boundedness of the sequence{u_n}n. From [29, Lemma 9], up to a subsequence, we have

⎧⎪

⎨

⎪⎩

u_nu inX^s₀(Ω)

u_n→u inL^p(Ω)for allp∈ 1,2^∗_s un→u a.e inR^N.

Using the Hölder inequality, it is straightforward to see that the sequence{un}nis also bounded in the spaceM(Ω), thus there exists two finite measuresμandνsuch that

(−Δ)^su_n^∗μ and |u_n|^2∗s∗ν inM(Ω)

From [25, Theorem 1.5], it follows that eitheru_n →uin L^2∗s(Ω)or there exist a setJ at most countable, two real sequences{μj}j∈J,{νj}j∈Jand distinct points{x_j}j∈J ⊂R^Nsuch that

ν= |u|^2∗s +

j∈J

νjδx_j (17)

and

μ=(−Δ)^su+ ˜μ+

j∈J

μjδxj (18)

for some positive finite measureμ˜, where

νj≤S_N,sμ²_j^∗s. (19)

Claim:the setJ is empty.

If not, there exists an index j₀such thatνj0 =0 atx_j0. Fixε >0 and consider a cut-off functionϑεsuch that

⎧⎪

⎨

⎪⎩

0≤ϑ_ε≤1 inΩ ϑε=1 inB(xj0, ε) ϑε=0 inΩ\B(x_j0,2ε).

Since the sequence{unϑε}nis still bounded inX^s₀(Ω), we have that

n→∞lim I_a,b(u_n)[u_nϑε]=0, thus

o(1)=I_a,b (un)[unϑ_ε]=

a+bun²

un,unϑ_εX^s₀(Ω)−

Ω|un|^2s∗ϑ_εd x

=

a+bu_n²

Qu_n(y)(u_n(x)−u_n(y))(ϑε(x)−ϑε(y))

|x−y|^N+2s d x d y +

Qϑε(x)(u_n(x)−u_n(y))²

|x−y|^N+2s d x d y

−

Ω|u_n|^2∗sϑεd x. (20) asn→ ∞. By using the Hölder inequality, we estimate the first term of (20)

a+bun²

Qun(y)(un(x)−un(y))(ϑ_ε(x)−ϑ_ε(y))

|x−y|^N+2s d x d y

≤

Q

(u_n(x)−u_n(y))²

|x−y|^N+2s d x d y

Qu²_n(y)(ϑε(x)−ϑε(y))²

|x−y|^N+2s d x d y C

Qu²_n(y)(ϑ_ε(x)−ϑ_ε(y))²

|x−y|^N+2s d x d y for someC >0. As in [3, Lemma 2.1], we have that

ε→0limlim sup

n→∞

Q|un(y)|²|ϑ_ε(x)−ϑ_ε(y)|²

|x−y|^N+2s d x d y=0. (21) Regarding the second term of (20), recalling (18), we get

n→∞lim

a+bun²

Qϑ_ε(x)(un(x)−u_n(y))²

|x−y|^N+2s d x d y

≥ lim

n→∞

a

R^2N\B(xj0,2ε)^c×Ω^cϑε(x)(un(x)−un(y))²

|x−y|^N+2s d x d y +b

Qϑε(x)(un(x)−un(y))²

|x−y|^N+2s d x d y 2

≥a

R^2N\B(xj0,2ε)^c×Ω^cϑε(x)(u(x)−u(y))²

|x−y|^N+2s d x d y+aμj0

+b

Qϑε(x)(u(x)−u(y))²

|x−y|^N+2s d x d y 2

+bμ²_j0. Hence

ε→0lim lim

n→∞

a+bun²

Qϑ_ε(x)(un(x)−u_n(y))²

|x−y|^N+2s d x d y≥aμj₀+bμ²_j0. (22) Finally, exploiting (17) we have

ε→0lim lim

n→∞

Ω|un|^2∗sϑεd x= lim

ε→0

Ω|u|^2∗sϑεd x+νj0 =νj0. (23) Putting together (21), (22) and (23), and using (19), we obtain

0≥aμj0+bμ²_j0−νj0 ≥aμj0+bμ²_j0−S⁻

2∗s 2 N,s μ²

∗s 2 j0 =μj0

a+bμj0−S⁻

2∗s N,s2μ²

∗s 2−1 j0

. We define

f˜_N,s(ζ )=a+bζ−S⁻

2∗s

N,s2ζ^22∗s−1 forζ ≥0.

At this point, noting that the conditiona^(N−4s)/2sb>PSN,simplies f˜_N,s(x) >0, we deduce a+bμj0−S⁻

2∗ 2s N,s μ²

∗s 2−1 j₀ >0. Henceμj0 =0, and recalling (19)νj0 =0 as well.

So the setJ = ∅, and using the Brezis-Lieb lemma (see [5, Theorem 1]) we can rewrite (17) as

nlim→∞

Ω|u_n|^2∗sd x=

Ω|u|^2∗sd x. Henceun →uinL^2∗s(Ω)and

n→∞lim

Ω|un|^2∗s−2u_n(u−u_n)d x=0. (24) Coupling (24) and the fact that_Ia,b(u_n)→0 asn→ ∞we get

0= lim

n→∞I_a,b (u_n)[u_n−u]= lim

n→∞

a+bu_n²

u_n,u_n−uX^s₀(Ω)

−

Ω|u_n|^2∗s−2u_n(u_n−u)d x

= lim

n→∞

a+bun²

un,un−uX₀^s(Ω).

From the last chain of equalities, recalling that{un}n⊂X^s₀(Ω)is bounded, we obtain

n→∞limun,u_n−uX₀^s(Ω)=0. (25)

To conclude the proof it suffices to notice that thanks to (25) andunuwe have un−u²= un,un−uX^s₀(Ω)− u,un−uX^s₀(Ω)→0

asn→ ∞.

Proof of Theorem1 (iii)In order to establish the convexity we will show that I_a,b (u)[v, v]≥0 for allu, v∈X^s₀(Ω).

Differentiating (7) we notice that

I_a,b (u)[v, v]=av²+bu²v²−(2^∗_s −1)

Ω|u|^2∗s−2v²d x. (26) Using the Hölder and the Sobolev inequalities we get

Ω|u|^2∗s−2v²d x≤ u²₂^∗s∗⁻²

s v²₂∗

s ≤S⁻

2∗s

N,s2u^2∗s−2v². (27) Putting together (26) and (27) we obtain

I_a,b (u)[v, v]≥ v²

a+bu²−(2^∗_s −1)S⁻

2∗ 2s

N,s u^2∗s−2

. At this point we set

fˆN,s(ζ )=a+bζ²−(2^∗_s−1)S²

∗s 2

N,sζ^2∗s−2 for allζ ≥0,

and we want to prove that it is positive on [0,∞). Indeed, with a simple computation it is possible to show that fˆN,sattains its global minimum at

ˆ m_N,s =

⎛

⎜⎝ 2bS

2∗s 2 N,s

(2^∗_s −1)(2^∗_s−2)

⎞

⎟⎠

1 2∗

s−4

and that

fˆN,s(ζ )≥0⇔a^N−4s2s b≥C_N,s

for allζ ≥0.

Remark 1 It is clear from the proof that the functionalIa,b is strictly convex provided that a^(N−4s)/2sb>CN,s.

3 Application to a perturbed Kirchhoff problem

This section is devoted to study an application of Theorem1. More precisely we want to study the set of solutions of the perturbed problem

⎧⎨

⎩

a+b

Q

|u(x)−u(y)|²

|x−y|^n+2s d x d y

(−Δ)^su= |u|^2∗s−2u+λg(x,u) inΩ

u=0 inR^N\Ω

(P_a,b^λ ) where as beforea,bare real positive parameter,Ωis a bounded domain andλ >0. As for g, we generalize to the fractional case the assumptions present in [13]. Namely, we make the following assumptions:

(H₁) g:Ω×R→Ris a Carathéodory function such thatg(x,0)=0 a.e. inΩ;

(H₂) g(x,t) >0 for everyt >0 andg(x,t) <0 for everyt <0 a.e. inΩ. In addition, we require that there is aμ >0 such thatg(x,t)≥μ >0 a.e inΩand for every t ∈I, whereI is some open interval of(0,∞);

(H3) there is a constantc>0 andp∈(2,2^∗_s)such thatg(x,t)≤c(1+ |t|^p−1)a.e. inΩ;

(H4) limt→0g(x,t)/|t| =0 uniformly with respect tox∈Ω.

Using a variational approach, we investigate the existence of critical points of the func- tional defined on the spaceX^s₀(Ω)

I_a^λ_,b(u):=a

2u²+b

4u⁴− 1 2^∗_su²₂^∗s∗

s −λ

ΩG(x,u)d x where we denote withG(x,t)=

_t

0

g(x, ω)dω.

We begin the treatment of our problem by proving a series of technical results that will be useful throughout this section.

Remark 2 Before starting, let us recall the functions

f_N,s(ζ ):= a 2+ b

4ζ²− S⁻

2∗ 2s N,s

2^∗_s ζ^2∗s−2 and

f˜_N,s(ζ )=a+bζ−S⁻

2∗s N,s2 ζ^22∗s−1

defined in the proofs of Theorems1(i)and1(ii). As we have already seen these functions have a unique local minimizer attained respectively at

mN,s = b

2 2^∗_s 2^∗_s−2S

2∗s N,s2

¹

2∗s−4

,

and

˜ m_N,s =

2b 2_s^∗−2S

2∗ 2s N,s

_2∗¹

s−4

.

Furthermore, fN,s(mN,s) > 0 if and only ifa^N−4s2s b > LN,s and fN,s(mN,s) = 0 when a^N−4s2s b = LN,s. Analogously f˜N,s(m˜N,s) > 0 if and only ifa^(N−4s)/2sb > PSN,s and

f˜N,s(m˜N,s)=0 whena^(N−4s)/2sb=PSN,s. Proposition 1 Let u∈X^s₀(Ω)\ {0}. We have that:

(i) for everyζ >0it holds a

2u²+b

4ζ²u⁴− 1

2^∗_sζ^2∗s−2> f_N,s(ζu)u²; (ii) for everyζ >0it holds

au²+bζ²u⁴− u²₂^∗s∗

sζ^2∗s−2 > f˜_N,s(ζu)u².