arXiv:2206.11873v1 [math.AP] 23 Jun 2022

arX

iv:2

206.

1187

3v1

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Jun

2022

A BOURGAIN-BREZIS-MIRONESCU FORMULA FOR

ANISOTROPIC FRACTIONAL SOBOLEV SPACES AND

APPLICATIONS TO ANISOTROPIC FRACTIONAL

DIFFERENTIAL EQUATIONS

IGNACIO CERESA DUSSEL AND JULIAN FERNANDEZ BONDER

Abstract. In this paper we prove Bourgain-Brezis-Mironescu’s type results(cf. [4]) (BBM for short) for an energy functional which is strongly related tothe fractional anisotropic p-Laplacian. We also provide with the analogous ofMaz’ya-Shaposhnikova (see [23]) type results for these energies and finally weapply these results to analyze the stability of solutions to anisotropic fractionalp−laplacian equations.

1. Introduction

The study of fractional order Sobolev spaces its a classical topic in analysis thathas drawn the attention of the mathematical community at least since the middleof the XX century. See, for instance the classical book of Stein [27].

In the last decades, there has been an increasing interests in this topic since newapplications of these spaces appear to some models in different fields of the naturalsciences such as physical models [12, 14, 17, 20, 24, 28], finance [1, 21, 26], fluiddynamics [8, 10], ecology [19, 22, 25] and image processing [18].

In particular, a relevant problem is the transition between nonlocal models andlocal ones. This, for instance, is the transition of the fractional space W s,p to theclassical Sobolev space W 1,p as the fractional parameter s ↑ 1 and also the casewhere the fractional parameter s ↓ 0 where the spaceW s,p approaches the Lebesguespace Lp. These problems were analyzed in the seminal papers [4] (for s ↑ 1) and[23] (for s ↓ 0).

An interesting application of the results in [4] is the asymptotic behavior ofsolutions to the fractional laplacian (or p−laplacian) as s ↑ 1. That is, if weconsider the problem

(1.1)

(−∆p)

su = f in Ω

u = 0 in Rn \ Ω,

where (−∆p)su is the fractional p−Laplace operator defined as

(−∆p)su(x). = p.v.(1− s)Kn,p

∫

Rn

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|n+spdy,

2020 Mathematics Subject Classification. 35J92, 35R11, 46E35.Key words and phrases. Fractional energies, fractional order Sobolev spaces, anisotropic

Sobolev spaces.

1

http://arxiv.org/abs/2206.11873v1

2 I. CERESA DUSSEL AND J. FERNANDEZ BONDER

and us is a solution to (1.1), then one expect that as s ↑ 1, us converge to a solutionof the local problem

−∆pu = f in Ω

u = 0 on ∂Ω.

This problem was analyzed, for instance, in [2, 3, 5, 15, 16], and was found to be

true in several cases, for instance if f = f(x) ∈ Lp′

(Ω) and if f = f(x, u) undersome growth control on f .

On the other hand, it is interesting to analyze Sobolev spaces and the corre-sponding operators in the case where some anisotropy is present. This study wascarried on in [7] where the authors analyze the fractional anisotropic Sobolev space,that in each coordinate direction the function has different fractional regularity anddifferent integrability. That is, in [7] the authors consider n different fractional pa-rameters s1, . . . , sn and n different integrability parameters p1, . . . , pn and definethe space of functions u(x) = u(x1, . . . , xn) such that

∫

Rn

∫

R

|u(x+ hei)− u(x)|pi

|h|1+sipidhdx <∞, for i = 1, . . . , n,

where ei = (0, . . . , 1︸︷︷︸i

, . . . , 0).

In the above mentioned paper [7] the authors study these fractional anisotropicSobolev spaces, prove some Sobolev-type immersion theorems and the existence ofextremals in R

n in the critical case.

In view of the above discussion, the purpose of this paper is to extend the classicalresults of [4] and [23] to the anisotropic case and then apply these extensions to thestudy of the asymptotic behavior of solutions to anisotropic fractional p−Laplaceequations when (some) fractional parameters approaches 1.

Organization of the paper. The paper is organized as follows. In Section 2, weintroduce the necessary preliminaries needed in this work, and discuss in detail someof the results in [7] that are related to our paper. Moreover we prove some impor-tant estimates on anisotropic fractional energies that will play a key role in the restof the article (see for instance Lemma 2.6). In Section 3 we prove the extension ofthe classical Bourgain-Brezis-Mironescu’s result [4] to the anisotropic case, see The-orems 3.1 and 3.2. In Section 4 we show the extension of Maz’ya-Shaposhnikova’sresult to our case, Theorem 4.1, and finally in Section 5 we analyze the applicationsof the above mentioned results to the asymptotic behavior of solutions to fractionalanisotropic p−Laplace problems when the fractional parameters (or some of them)goes to 1. We analyze the cases where the source term f is a fixed function of thespace variable x (Corollary 5.4), then the general case where the source term is anonlinearity of the form f = f(x, u) (Theorem 5.5) and finally, we study the caseof ground state solution proving that the limit of ground state solutions to frac-tional problems converge to the corresponding ground state solutions of the limitproblems (Theorem 5.11).

2. Preliminaries

Let us start by defining the anisotropic spaces that will be considered in thiswork.

A BBM FORMULA FOR ANISOTROPIC FRACTIONAL SOBOLEV SPACES 3

Given i = 1, . . . , n, s ∈ (0, 1] and p ∈ (1,∞) we define the fractional ith−Sobolevspace as

W s,pi (Rn) = u ∈ Lp(Rn) such that J i

s,p(u) <∞,

where

J is,p(u) =

(1− s)s

∫

Rn

∫

R

|u(x+ hei)− u(x)|p

|h|1+spdhdx if s < 1

2

p

∫

Rn

|∂xiu|p dx if s = 1.

It is easy to see that W s,pi (Rn) is a Banach space with norm

‖u‖i,s,p = ‖u‖p + (J is,p(u))

1/p,

which is separable and reflexive.This space consists on Lp functions that has some mild regularity in the ith−variable.

In order to control every variable, we proceed as follows. Let s = (s1, . . . , sn) andp = (p1, . . . , pn), with si ∈ (0, 1] and pi ∈ (1,∞) for i = 1, . . . , n. Then define

W s,p(Rn) =n⋂

i=1

W si,pi

i (Rn).

This anisotropic Sobolev space is then a separable and reflexive Banach space withnorm

‖u‖s,p =

n∑

i=1

‖u‖i,si,pi.

In most applications, what turns out to be more useful than the norm is thefunctional Js,p that is defined as

Js,p(u) =

n∑

i=1

1

piJ is,p(u)

for u ∈ W s,p(Rn).If we consider Ω ⊂ R

n, then we define the anisotropic Sobolev space of functionsvanishing at the boundary ∂Ω as

W s,p0 (Ω) = u ∈W s,p(Rn) : u|Ωc = 0.

Remark 2.1. Recall that this is not the most common definition of functions van-ishing at the boundary. An alternative definition is

(2.1) W s,p0 (Ω) = C∞

c (Ω),

where the closure is taken with respect to the norm ‖ · ‖s,p.In the classical case, where s = s1 = s2 = · · · = sn and p = p1 = p2 = · · · = pn,

it is proved in [11] that both spaces agree, W s,p0 (Ω) = W s,p

0 (Ω), if for instance, Ωis Lipschitz.

We don’t investigate this issue here for the anisotropic case, and will considerthe definition given in (2.1).

Observe that if we define

pmin = minp1, . . . , pn and pmax = maxp1, . . . , pn

then, by standard interpolation we have that

W s,p(Rn) = u ∈ Lpmin(Rn) ∩ Lpmax(Rn) : Js,p(u) <∞ .


In the local anisotropic case, i.e. when s1 = s2 = · · · = sn = 1, it is proven in[13] that the space W 1,p(Rn) is continuously embedded in Lp

∗

(Rn), where p∗ isthe associated critical exponent given by

p∗ =n∑n

i=1 p−1i − 1

.

On the other hand, in the general fractional anisotropic case, Sobolev immersiontheorems for these spaces was studied in [7]. In that paper, the authors define thequantities

s =

(1

n

n∑

i=1

1

si

)−1

, sp =

(1

n

n∑

i=1

1

sipi

)−1

.

Then, assuming that sp < n the following critical Sobolev exponent is introduced

(2.2) p∗s = p∗ =

nsp/s

n− sp.

Under the further assumption that pmax < p∗ the following Sobolev-Poincare in-equality is proved in [7]:

0 < S = infu∈W s,p(Rn)

‖u‖p∗=1

Js,p(u).

where the constant S depends on n,p∗,p∗ − pmax and on a lower bound s0 of sifor i = 1, . . . , n.

Moreover, a compact-embedding theorem, W s,p(Rn) ⊂⊂ Lqloc(R

n), for every1 ≤ q < p∗ is also proved in [7].

In the remaining of the section we prove some results on the functionals J is,p that

will be helpful in the sequel. When dealing with a single functional, without lossof generality we can assume that i = 1.

Lemma 2.2. Let u be a function in W 1,p1 (Rn), then

J1s,p(u) ≤

1

p

(s ‖∂x1u‖

pp + (1 − s)2p+1‖u‖pp

).

Proof. Given u ∈ C1c (R

n) it is a well known fact that∫

Rn

|u(x+ he1)− u(x)|p dx ≤ |h|p‖∂x1u‖pp,

so∫

R

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+spdxdh =

∫ 1

−1

1

|h|1+sp

(∫

Rn

|u(x+ he1)− u(x)|p dx

)dh

+

∫

|h|>1

1

|h|1+sp

(∫

Rn

|u(x+ he1)− u(x)|p dx

)dh

=I + II.

For the first term,

(2.3) I ≤ ‖∂x1u‖pp 2

∫ 1

0

hp

|h|1+spdh =

1

p(1− s)‖∂x1u‖

pp

and for the second term,

(2.4) II ≤ 2p+1‖u‖pp

∫ ∞

1

1

|h|1+spdh = 2p+1||u||pp

1

sp.


Now,

I ≤ ‖∂x1ηk‖p∞

∫

|h|<1

1

|h|1+sp−pdh =

2p+1

kp1

p(1− s).

Finally,

II ≤ 2p−1‖ηk‖p∞

∫

|h|≥1

1

|h|1+spdh =

2p

sp.

and our lemma has been proved.

The next lemma is proved in [4] and is crucial in the arguments that follows.

Lemma 2.5. Let φ, ψ : (0, 1) → R be measurables functions and suppose thatφ(2h) ≤ φ(h) for h ∈ (0, 1) and that ψ is nonincreasing.Then given r > −1,

∫ 1

0

hrφ(h)ψ(h) dh ≥r + 1

2r+1

∫ 1

0

hrφ(h) dh

∫ 1

0

hrψ(h) dh.

With the help of Lemma 2.5 we can prove the following immersion result.

Lemma 2.6. Let 1 < p <∞ and 0 < s1 < s2 < 1 then

J1s1,p(u) ≤ 2(1−s1)pJ1

s2,p(u) +(1 − s1)2

p+1

p‖u‖pp,

for all u ∈ Lp(Rn).

Proof. First, let us begin by observing that∫

R

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+spdxdh = 2

∫ ∞

0

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+spdxdh.

Hence, the lemma will be proved if we show that

(1− s1)

∫ ∞

0

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+s1pdxdh

≤2(1−s1)p(1− s2)

∫ ∞

0

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+s2pdxdh+

(1− s1)2p

s1p‖u‖pp,

Define F as

F (h) =

∫

Rn

|u(x+ he1)− u(x)|p dx,

since

F (2h) ≤ 2p−1

(∫

Rn

|u(x+ 2he1)− u(x+ he1)|p dxdh

)

+ 2p−1

(∫

Rn

|u(x+ he1)− u(x)|p dxdh

)

= 2p∫

Rn

|u(x+ he1)− u(x)|p dxdh = 2pF (h),

We can take g(h) = F (h)/hp and get that g(2h) ≤ g(h) for all h > 0.On the other hand,

∫ 1

0

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+spdxdh =

∫ 1

0

F (h)

|h|1+spdh

=

∫ 1

0

g(h)

|h|1−(1−s)pdh

(2.5)


Now, let us consider 0 < s1 < s2 < 1, then∫ 1

0

g(h)

|h|1−(1−s2)pdh =

∫ 1

0

1

|h|1−(1−s1)pg(h)

1

|h|(s2−s1)pdh

and applying Lemma 2.5 with r = (1− s1)p− 1 and ψ(h) = h−(s2−s1)p, we obtain∫ 1

0

g(h)

|h|1−(1−s2)pdh ≥

(1 − s1)p

2(1−s1)p

∫ 1

0

g(h)

h1−(1−s1)pdh

∫ 1

0

1

h1−(1−s2)pdh

=1

2(1−s1)p

1− s11− s2

∫ 1

0

g(h)

h1−(1−s1)pdh.

Recalling (2.5)

1− s12(1−s1)p

∫ 1

0

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+s1pdxdh

≤ (1− s2)

∫ 1

0

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+s2pdxdh.

Finally, ∫ ∞

1

∫

Rn

|u(x+ he1)− u(x)|p

|h|1+s1pdxdh ≤

2p‖u‖pps1p

this completes the proof.

3. BBM-type results for W s,pi

In this section we extend the results in the seminal paper [4] to the anisotropiccase. Our first result conserns with the asymptotic behavior of the funcional J1

s,p

as s ↑ 1 in a pointwise sense.

Theorem 3.1. Let u ∈ Lp(Rn), s ∈ (0, 1) then

lims→1

J1s,p(u) = J1

1,p(u).

Proof. To prove the theorem 3.1 we split the proof in 3 steps.

Step 1: Let u ∈ C20 (R

n) , s ∈ (0, 1) and p ∈ (1,∞). Arguing as in [4, Theorem2] we easily obtain that

∣∣∣∣|u(x+ he1)− u(x)|p

|h|p− |∂xi

u|p∣∣∣∣ ≤ C|h|,

where C depends on the C2−norm of u and p.Naming

A =|u(x+ he1)− u(x)|p

|h|p− |∂xi

u|p

and taking the integral over R,

∫

R

Adh =

∫

|h|≤1

Adh+

∫

|h|>1

Adh,

where by routine integration,∫

|h|≤1

Adx = 2C1

1 + p(1− s)and

∫

|h|>1

Adx = 2p‖u‖p∞ps

.


Finally, taking limit s→ 1 we conclude that

(3.1) lims→1

s(1− s)

∫

R

|u(x+ he1)− u(x)|p

|h|1+spdh =

2

p|∂x1u|

p.

Now, the key is look up an integrable majorant for (3.1).Let

Us(x) =

∫

R

|u(x+ he1)− u(x)|p

|h|1+spdh.

Then since u ∈ C20 (R

n), there exists R > 0 such that supp(u) ⊂ QR, whereQR = [−R,R]n.

Thus, we split the problem:

• If |x1| ≤ 2R

|Us(x)| =

∫

R

|u(x+ he1)− u(x)|p

|h|1+spdh

=

(∫

|h|≤3R

+

∫

|h|>3R

)|u(x+ he1)− u(x)|p

|h|1+spdh

,

where we bound the first integral by

∫

|h|≤3R

|u(x+ he1)− u(x)|p

|h|1+spdh ≤ ||∂x1u||

p∞

3Rp

p(1− s),

and the second by∫

|h|>3R

|u(x+ he1)− u(x)|p

|h|1+spdh ≤ ||u||p∞

3R−p(1/2)

(1− s)p.

Then

|s(1− s)Us(x)| ≤ CχQ2R(x),

where C is a constant depending of p,R but not on s.• If |x1| > 2R,

|Us(x)| =

∫

R

|u(x1 + h, x′)|p

|h|1+spdh ≤

(2

|x1|

)1+sp ∫

R

|u(x1, x′)|pdx1.

Where we have used that |h| ≥ |x1| − |x1 + h| ≥ |x1| −R ≥ 12 |x1|.

Observe that

G(x′) :=

∫

R

|u(x1, x′)|pdx1 ∈ L1(Rn−1)

Therefore,

|s(1− s)Us(x)| ≤ C

(χQ2R(x) + χQc

2R(x)

(2

|x1|

)1+(1/2)p

G(x′)

)∈ L1(Rn).

To conclude this step, we only have to use the Lebesgue’s Dominated ConvergenceTheorem and obtain

(3.2) lims→1

s(1− s)

∫

Rn

∫

R

|u(x+ he1)− u(x)|p

|h|1+spdhdx =

2

p

∫

Rn

|∂x1u|pdx,

for all u ∈ C20 (R

n).


Step 2: Let u ∈ W 1,p1 (Rn), s ∈ (0, 1) and p ∈ (1,∞) . Take a sequence

ukk∈N ⊂ C20 (R

n) such that uk → u in Lp(Rn) and ∂x1uk → ∂x1u in Lp(Rn). Let

(3.3) [u]1,s,p = (J1s,p(u))

1/p, 0 < s ≤ 1.

Observe that [ · ]1,s,p defined in (3.3) is a seminorm and thus satisfies the triangleinequality. So

|[u]1,s,p − [u]1,1,p| ≤ |[u]1,s,p − [uk]1,s,p|+ |[uk]1,s,p − [uk]1,1,p|+ |[uk]1,1,p − [u]1,1,p|

= I + II + III.

Using lemma 2.2,

I ≤ |[uk − u]1,s,p| ≤ C(n, p) (‖uk − u‖p + ‖∂x1uk − ∂x1u‖p) .

which converges to zero if k → ∞, uniformly in s. The third term is bounded inmuch the same way.

Finally, the second term is consequence of Step 1, and so the proof of Step 2 iscomplete.

Step 3: Given u ∈ Lp(Rn) and assume that

(3.4) lim infs→1

J1s,p(u) <∞,

Let uk,ǫ ⊂ C∞0 (Rn) be given by

uk,ǫ = ρǫ ∗ (uηk)

and applying lemmas 2.3 and 2.4 together with (3.4), we obtain

lim infs→1

J1s,p(uk,ǫ) ≤ C,

where C is independent from k and ǫ > 0.Next, for any fixed k and ǫ, we apply Step 1 and obtain

C ≥ lims→1

J1s,p(uk,ǫ) = J1

1,p(uk,ǫ).

Hence, there exist a subsequence uj = ukj,ǫj ⊂ uk,ǫ such that ∂x1uj ∂x1u

weakly in Lp(Rn). Thus u ∈ W 1,p1 (Rn). and so the proof of Step 3 follows from

Step 2.This completes the proof of the Theorem.

Next, we will consider the case that, instead of having a fixed function u forevery s, the function also depends on the fractional parameter s.

Theorem 3.2. Let 0 < sk → 1 when k → ∞. Assume that uk ⊂ Lp(Rn) is suchthat

(3.5) supk

‖uk‖p <∞, uk → u in Lploc

(Rn) and supk∈N

J1sk,p

(uk) <∞.

Then,

(3.6) J11,p(u) ≤ lim inf

sk→1J1sk,p

(uk).


Proof. Let uk ⊂ Lp(Rn) be such that (3.5) holds. Let now t ∈ (0, 1) be fixed butarbitrary. By the Lp

loc(Rn) convergence of the sequence uk, we can assume that

uk → u a.e in Rn.

By Fatou’s lemma,

J1t,p(u) ≤ lim inf

k→∞J1t,p(uk)

Applying now Lemma 2.6, with M = supk ‖uk‖p we get

J1t,p(u) ≤ 2(1−t)p lim inf

kJ1sk,p

(uk) + (1− t)2p+1

pMp,

and the result follows taking t→ 1 and using Theorem 3.1.

Another useful result is the following compactness result for a sequence uk ∈W sk,p(Rn).

Theorem 3.3. Let p = (p1, . . . , pn) with pi ∈ (1,∞) for every i = 1, . . . , n. As-sume that ski → 1 as k → ∞ for i = 1, . . . , j and let sk = (sk1 , . . . , s

kj , sj+1, . . . , sn).

Then sk → s0 = (1, . . . , 1︸︷︷︸j

, sj+1, . . . , sn).

Let uk ⊂ Lpmin(Rn) ∩ Lpmax(Rn) be a sequence such that

supk∈N

‖uk‖pmax+ ‖uk‖pmin

<∞ supk∈N

Jsk,p(uk) <∞.

Then, if pmax < p∗s0, there exist a subsequence ukj

j∈N ⊂ ukk∈N such thatukj

→ u as j → ∞ in Lpmax

loc(Rn).

Proof. Since sk → s0, it follows that p∗sk

→ p∗s0. Therefore, we can assume that

pmax < p∗sk.

Now, take t ∈ (0, 1) close to 1, such that it t = (t, . . . , t︸︷︷︸j

, sj+1, . . . , sn), then

pmax < p∗t .

Applying now Lemma 2.6 we obtain that

Jt,p(uk) ≤ C(Jsk,p(uk) + 1) ≤ C for every k ∈ N.

Hence, we can apply the Rellich-Kondrashov type result for the space W t,p(Rn)[7, Theorem 2.1] and hence the convergence of ukk∈N (up to a subsequence) inLpmax

loc (Rn) is guaranteed.

To conclude the section, let us now make the connection of Theorems 3.1 and3.2 with the notion of Gamma-convergence.

Recall that in a metric space (X, d), a sequence of functions fk : X → R is saidto Gamma converges to a limit function f it this two conditions are satisfied:

• (lim inf inequalitty) For every x ∈ X and every sequence xkk∈N ⊂ Xsuch that xk → x, it holds that

f(x) ≤ lim infk→∞

fk(xk).

• (lim sup inequality) For every x ∈ X there exists a sequence (called therecovery sequence) ykk∈N ⊂ X such that

f(x) ≥ lim supk→∞

fk(yk).


The notion of Gamma convergence is a useful tool in the Calculus of Variationsand was introduced by De Giorgi in the 70’s. See [9] for a throughout introductionto the subject.

Observe that Theorems 3.1 and 3.2 immediately imply that J1s,p Gamma con-

verges to J11,p as s ↑ 1 in Lp(Ω) for every Ω ⊂ R

n bounded. We collect this statementin the following corollary.

Corollary 3.4. Let sk → 1 and p ∈ (1,∞). Then J1sk,p

Gamma converges to J11,p

as k → ∞ in Lploc

(Rn).

Proof. Just observe that Theorem 3.2 is exactly the lim inf inequality and for thelim sup inequality one just have to take the constant sequence uk = u as a recoverysequence and apply Theorem 3.1.

In the forthcoming application (cf. Section 5) it will be important to have theGamma convergence of the full functional

Js,p(u) =

n∑

i=1

1

piJ isi,pi

(u).

Moreover we consider a sequence sk = (sk1 , sk2 , . . . , s

kn) where for instance s

k1 , . . . , s

kj →

1 as k → ∞ and skj+1, . . . , skn remains constant.

Hence sk → s0 = (1, . . . , 1︸︷︷︸j

, sj+1, . . . , sn).

In principle, it is not true that sums of Gamma convergent function is Gammaconvergent (see [9]). However, in this case since the recovery sequence is the con-stant sequence (and in particular is the same for every function), the Gamma con-vergence of Jsk,p holds true.

Let us collect these results in the next corollary.

Corollary 3.5. Let p = (p1, . . . , pn) with pi ∈ (1,∞) for every i = 1, . . . , n. Letski → 1 as k → ∞ for i = 1, . . . , j and let sk = (sk1 , . . . , s

kj , sj+1, . . . , sn). Let

s0 = (1, . . . , 1︸︷︷︸j

, sj+1, . . . , sn). Then Jsk,p Gamma converges to Js0,p in Lpmax

loc(Rn).

Proof. The lim inf inequality for Jsk,p follows from the lim inf inequality for everyJ iski,pi

(i = 1, . . . , j) and the continuity of J isi,pi

for i = j + 1, . . . , n.

The lim sup inequality follows since the constant sequence uk = u is a recoverysequence for every J i

ski,pi

, i = 1, . . . , j.

4. Mazya-Shaposhnikova–type results

In this section we work on an adaptation of the seminal result of Maz’ya andShaposhnikova, [23]. Namely we prove the following theorem.

Theorem 4.1. Let u ∈ Lp(Rn) and assume that there exists s0 ∈ (0, 1) such thatu ∈ W s0,p

i (Rn). Then

lims→0

J is,p(u) = 2p−1‖u‖pp.

Proof. In [23], It is proved that, for any n ∈ N,

(4.1) lims→0

s

∫

Rn

∫

Rn

|u(x)− u(y)|p

|x− y|n+spdxdy = 2p−1|Sn−1|‖u‖pLp(Rn).


for all u ∈W s0,p(Rn) for some s0 > 0.In particular, if we apply this result in dimension 1 to v(x1) = u(x1, x

′) wherex′ ∈ R

n−1 is fixed, we obtain that

lims→0

s

∫

R

∫

R

|v(x1 + h)− v(x1)|p

|h|1+spdx1dh =

4

p

∫

R

|v(x1)|p dx1,

which in terms of u reads

(4.2) lims→0

s

∫

R

∫

R

|u(x1 + h, x′)− u(x1, x′)|p

|h|1+spdx1dh =

4

p

∫

R

|u(x1, x′)|p dx1,

So, to conclude our result we need to integrate with respect to x′ ∈ Rn−1 and

for that it remains to find an integrable majorant for

Vs(x′) := s

∫

R

∫

R

|u(x1 + h, x′)− u(x1, x′)|p

|h|1+spdx1dh.

For that purpose, we proceed as follows

Vs(x′) = s

(∫

R

∫

|h|≤1

+

∫

R

∫

|h|≥1

)|u(x1 + h, x′)− u(x1, x

′)|p

|h|1+spdx1dh = I + II.

For the first term

I = s

∫

R

∫

|h|≤1

|u(x1 + h, x′)− u(x1, x′)|p

|h|1+spdx1dh

≤ s0

∫

R

∫

|h|≤1

|u(x1 + h, x′)− u(x1, x′)|p

|h|1+s0pdx1dh

The inequality came from s ≤ s0, and thanks to u ∈ W s0,p(Rn) we conclude theintegrability of this part. For the second term

II = s

∫

R

∫

|h|≥1

|u(x1 + h, x′)− u(x1, x′)|p

|h|1+spdx1dh

≤ s

∫

R

2p|u(x1, x′)|p∫

|h|≥1

1

|h|1+spdhdx1

≤2p

p

∫

R

|u(x1, x′)|p dx1

and this last term is integrable in Rn−1 since u ∈ Lp(Rn).

These two bounds conclude the proof.

5. Applications to partial differential equations.

In this section, we will make use of the results in Section 3 to find connectionsbetween nonlocal pseudo p−Laplace operators and their local counterparts.

First, we will give some definitions and enforce notations that will be helpful.

Definition 5.1. Let Ω ⊂ Rn be a bounded subset. The topological dual spaces of

W s,p0 (Ω),W s,p

i,0 (Ω) will be denoted by W−s,p′

(Ω) and W−s,p′

i (Ω) respectively.

Now, it is easy to see that the functionals J is,p are Frechet differentiable for every

i = 1, . . . , n, every s ∈ (0, 1] and every p ∈ (1,∞). Even more (J is,p)

′ : W s,pi,0 (Ω) →


W−s,p′

i (Ω) is continuous and is given by

〈(J is,p)

′(u), v〉 =

s(1− s)p

∫

Rn

∫

R

|u(x+ hei)− u(x)|p−2(u(x+ hei)− u(x))(v(x + hei)− v(x))

|h|1+spdhdx,

if 0 < s < 1 and

〈(J i1,p)

′(u), v〉 = 2

∫

Rn

|∂xiu|p−2∂xi

u ∂xiv dx,

where 〈·, ·〉 denotes the duality pairing between W s,pi,0 (Ω) and its dual W−s,p′

i (Ω).

5.1. Pseudo fractional p−Laplace operator. Let si ∈ (0, 1], i = 1, . . . , n befractional parameters and define s = (s1, . . . , sn). Next, let pi ∈ (1,∞), i = 1, . . . , nbe integrable parameters and define p = (p1, . . . , pn).

Recall that the operator Js,p : Ws,p0 (Ω) → R that was defined as Js,p :=

∑ni=1

1piJ isi,pi

,

is then Frechet differentiable with derivative given by

J ′s,p : W

s,p0 (Ω) →W−s,p′

(Ω)

J ′s,p(u) =

n∑

i=1

1

pi(J i

si,pi)′(u).

So, we define the pseudo anisotropic p−Laplace operator as

(−∆p)su := (Js,p)

′(u).

Observe that in the case where s = (1, . . . , 1) (that is the pure local case) andp = (p, . . . , p), then

(−∆p)su :=

n∑

i=1

∂xi

(|∂xi

u|p−2∂xiu),

that is the well-known pseudo p-Laplace operator.Therefore, we want to study the equation

(5.1)

(−∆p)

su = f in Ω

u = 0 in Rn \ Ω

and, we say that u ∈ W s,p(Ω) is a weak solution of (5.1) if

〈(−∆p)su, v〉 =

∫

Ω

fv dx

for all v ∈ W s,p0 (Ω).

In order for a weak solution to be well defined, as usual we need to impose someintegrability conditions on the source term f .

We consider first the case where f = f(x) and then, the Sobolev immersion

theorem proved in [7] requires that f ∈ L(p∗)′(Ω).Existence and uniqueness of weak solutions to (5.1) is then a direct consequence

of the direct method in the calculus of variations, since the solution is the uniqueminimizer of the functional

I(v) := Js,p(v) −

∫

Ω

fv dx,

which is a strictly convex, coercive and continuous functional in W s,p0 (Ω).


Let us summarize all of this facts into a single statement.

Proposition 5.2. Let Ω ⊂ Rn be a bounded open set, let s = (s1, . . . , sn), si ∈

(0, 1), p = (p1, . . . , pn), pi ∈ (1,∞) and p∗ the critical Sobolev exponent defined in(2.2). Assume that pmax < p∗.

Then, for every f ∈ L(p∗)′(Ω), there exists a unique weak solution u ∈ W s,p0 (Ω)

to (5.1).

5.2. Asymptotic behavior of solutions. The objective of this section is to an-alyze the asymptotic behavior of the solution to (5.1) when (some of the) si → 1.

To this end assume that sk1 , . . . , skj → 1 (as k → ∞) and sj+1, . . . , sn remain

fixed and consider sk := (sk1 , . . . , skj , sj+1, . . . , sn) and the functionals

Ik(v) := Jsk,p(v)−

∫

Ω

fv dx.

By Proposition 5.2, there exists a unique minimizer uk of Ik in W sk,p0 (Ω). Hence

we want to study the behavior of the sequence ukk∈N as k → ∞.To this end, let us define s0 := (1, . . . , 1︸︷︷︸

j

, sj+1, . . . , sn) and

I0(v) := Js0,p(v)−

∫

Ω

fv dx,

where

Js0,p(v) :=

j∑

i=1

1

piJ i1,pi

(v) +

n∑

i=j+1

1

piJ isi,pi

(v).

Next we make full use of the results in Section 3 and conclude the followingtheorem:

Theorem 5.3. With the preceding notation, the functionals Ik : Lpmin(Ω) → R

defined as

Ik(v) =

Ik(v) if v ∈W sk,p

0 (Ω)

∞ elsewhere

Gamma-converges to I : Lpmin(Ω) → R defined as

I(v) =

I(v) if v ∈ W s0,p

0 (Ω)

∞ elsewhere

Proof. By Corollary 3.5 we have that Jsk,p Gamma converge to Js0,p as k → ∞.So the result follows since the functional u 7→

∫Ωfu dx is continuous in Lpmin(Ω)

(see [9]).

An immediate consequence of this result is the convergence of the solutions of(5.1) to the corresponding limiting problem.

Corollary 5.4. Let uk ∈ W sk,p0 (Ω) be the weak solution to

(−∆p)

skuk = f in Ω

uk = 0 in Rn \ Ω,


then uk → u0 in Lpmin(Ω) where u0 ∈W s0,p0 (Ω) is the weak solution to

(−∆p)

s0u0 = f in Ω

u0 = 0 in Rn \ Ω,

5.3. The semilinear-like case. In this subsection, we investigate the case wherethe source term f depends also on the solution itself u. That is f = f(x, u).

In this case, in order to have the notion of weak solution some conditions on fhas to be imposed. Namely:

(f1) f : Ω → R× R is Caratheodory function.(f2) There exist a constant C > 0 and an exponent q ≤ p∗ such that

|f(x, z)| ≤ C(1 + |z|q−1)

Recall that (f1) implies that if f(x, u(x)) is measurable whenever u(x) is mea-

surable. Moreover, (f2) implies that if u ∈ Lp∗

(Ω), then f(x, u(x)) ∈ L(p∗)′(Ω).Hence, we say that u is a weak solution to

(5.2)

(−∆p)

su = f(x, u) in Ω

u = 0 in Rn \ Ω,

if u ∈ W s,p0 (Ω) and

〈J ′s,p(u), v〉 =

∫

Ω

f(x, u(x))v(x) dx, for every v ∈W s,p0 (Ω).

Under these conditions, we can prove a stability result for weak solutions to(5.2).

Theorem 5.5. Assume that for each k ∈ N, there exists uk ∈ W sk,p0 (Ω) a weak

solution to

(5.3)

(−∆p)

skuk = f(x, uk) in Ω

uk = 0 in Rn \ Ω,

Moreover, assume that the sequence ukk∈N is precompact in Lpmin(Ω), boundedin Lq(Ω) and that sk → s0 as k → ∞ and finally assume that the exponent q in(f2) satisfy that q < p∗

s0. Hence, any accumulation point u of the sequence ukk∈N

belongs to W s0,p0 (Ω) and it is a weak solution to

(−∆p)

s0u = f(x, u) in Ω

u = 0 in Rn \Ω,

In order to prove Theorem 5.5, a couple of lemmas, that are straightforwardadaptation of [15, Lemmas 2.7 and 2.8].

Lemma 5.6. With the same assumptions and notations as in Theorem 5.5, letu ∈ W s0,p

0 (Ω) be fixed and for any k ∈ N, let vk ∈ W sk,p0 (Ω) be such that

supk∈NJsk,p(vk) <∞. Then, for t > 0,

Jsk,p(u+ tvk) ≤ Jsk,p(u) + t〈(−∆p)sku, vk〉+ o(t),

where o(t) is uniform in k ∈ N.

Lemma 5.7. Under the same assumptions that Lemma 5.6, if moreover vk → vin Lpmin(Ω). Then, for every u ∈W s0,p

0 (Ω) we have

〈(Jsk,p)′(u), vk〉 → 〈(Js0,p)

′(u), v〉.


With the help of these two lemmas, we can now proceed with the proof ofTheorem 5.5.

Proof of Theorem 5.5. First observe that

〈(Jsk,p)′(uk), uk〉 ≥ pminJsk,p(uk).

Hence, using (f2), and the fact that uk are weak solutions to (5.3), we get that

Jsk,p(uk) ≤ C(1 + ‖uk‖qq).

Therefore, since ukk∈N is bounded in Lq(Ω), we have that

(5.4) supk∈N

Jsk,p(uk) <∞.

Also, since uk → u in Lpmin(Ω), passing to a subsequence, if necessary, we get thatuk → u a.e. in Ω and, moreover, using Theorem 3.3, we obtain that u ∈W s0,p

0 (Ω).

Next, we define ηk = f(x, uk(x)) = (−∆p)skuk ∈ W−sk,p

′

(Ω). Let us see that

ηk is bounded in W−s0,p′

(Ω).First, notice that p∗

sk→ p∗

s0as k → ∞ and q < p∗

s0, then (q − 1)(p∗

s0)′ < p∗

sk,

for k large. Then

〈ηk, v〉 =

∫

Ω

f(x, uk)v dx

≤ C

∫

Ω

(1 + |uk|q−1)|v| dx ≤ C

∫

Ω

|v| dx+ C

∫

Ω

|uk|q−1|v| dx

≤ C‖v‖p∗s0+ C‖v‖p∗

s0

(∫

Ω

|uk|(q−1)(p∗

s0)′ dx

)1/(p∗s0

)′

.

Since (q − 1)(p∗s0)′ < p∗

sk, by the Sobolev immersion for the fractional anisotropic

spaces proved in [7] and the uniform bound on Jsk,p(uk), (5.4), we obtain that

‖uk‖(q−1)(p∗s0

)′ ≤ C, for every k ∈ N.

Hence, ηk is bounded in W−s0,p′

(Ω).

Therefore, up to some subsequence, there exists η ∈W−s0,p′

(Ω) such that ηk η

weakly in W−s0,p′

(Ω).Since uk is a weak solution of (5.3), for any v ∈W s0,p

0 (Ω),

0 = 〈ηk, v〉 −

∫

Ω

f(x, uk)vdx

taking limit as k → ∞ the first term converge weakly, while the second termconverge by the dominated convergence theorem and [6, Theorem 4.9],

0 = 〈η, v〉 −

∫

Ω

f(x, u)vdx.

Now, we want to identify η, let us see that

〈η, v〉 = 〈(−∆p)s0u, v〉.

For that purpose, we will use the monotonicity of the operator and the fact thatuk is a weak solution of (5.3). In fact,

0 ≤ 〈(−∆p)skuk, uk − v〉 − 〈(−∆p)

skv, uk − v〉

=

∫

Ω

f(x, uk)(uk − v)dx − 〈(−∆p)skv, uk − v〉


Taking limit k → ∞ and for Lemma 5.7,

0 ≤

∫

Ω

f(x, u)(u− v)dx − 〈(−∆p)s0v, u − v〉

If we take v = u− tw, w ∈W s0,p(Ω) given and t > 0 , we obtain that

0 ≤ 〈η, w〉 − 〈−∆p)s0(u− tw), w〉.

Finally, taking t→ 0

0 ≤ 〈η, w〉 − 〈−∆p)s0(u), w〉.

The proof is now completed.

5.4. Ground-state solutions. The purpose of this subsection is to investigatefurther the results of the preceding one and analyze the case where the sequence ofsolutions ukk∈N are ground-state solutions to the problem:

(5.5)

(−∆p)

skuk = f(x, uk) in Ω

uk = 0 in Rn \ Ω,

and its limit behavior to

(5.6)

(−∆p)

s0u = f(x, u) in Ω

u = 0 in Rn \ Ω,

Recall that ground state solutions are minimizers of the energy functional

Is,p(u) = Js,p(u)−

∫

Ω

F (x, u) dx,

restricted to the Nehari manifold

Ns,p = v ∈ W s,p0 (Ω) \ 0 : 〈(Is,p)

′(v), v〉 = 0.

where F (x, z) =∫ z

0f(x, t) dt is the primitive of f.

On the nonlinearity f , besides (f1) and (f2), it will be assumed to fulfill thefollowing further structural hypothesis that are standard when consider groundstate solutions in nonlinear problems

(f3)F (x, z)

|z|pmax→ ∞ as |z| → ∞ uniformly with respect to x ∈ Ω.

(f4) f(x, z) = (|z|pmax−1) as z → 0 uniformly with respect to x ∈ Ω.(f5) For almost every x ∈ Ω

f(x, z)

|z|pmax−1is strictly increasing on (−∞, 0) ∪ (0,∞)

(f6) There is a µ > pmax such that

µF (x, z) ≤ zf(x, z)

Also, as in the preceding subsection, the exponent q in (f2) must be subcritical, i.e.q < p∗.

It is well known that under hypotheses (f1)-(f6) a ground state actually exists.In fact, in our context, a ground state solution, us, is a mountain pass solution.


Remark 5.8. Observe that the Nehari manifold Ns,p has the property that givenany nonzero function u ∈W s,p

0 (Ω), there exists a unique t > 0 such that tu ∈ Ns,p.We denote this scalar as tsu.

The existence and uniqueness of this scalar tsu, follows by using standard argu-ments in the Calculus of Variations. That is, given t ∈ R and u ∈ W s,p

0 (Ω) definethe function

φ(t) = Js,p(tu)−

∫

Ω

F (x, tu) dx.

On the one hand, it is easy to verify that φ(0) = 0, φ′(t) ≥ 0 for small values oft > 0 (these follows from the definition of φ and (f4)). Moreover, by (f3) we obtainthat φ(t) → −∞ when t→ ∞. So we conclude the existence of at least one criticalpoint of φ.

On the other hand, φ′(t) verifies that

φ′(t) =

n∑

i=1

tpiJ isi,pi

(u)−

∫

Ω

f(x, ut)tu dx

= tpmax

[C +

∑

pi 6=pmax

tpi−pmaxJ isi,pi

(u)−

∫

Ω

f(x, ut)upmax

|ut|pmax−1dx

︸︷︷︸A

].

By (f5), A is strictly decreasing, so the critical point must be unique.

By the previous remark, it is easy to check that a ground state solution mustsatisfy

cs = Is,p(us) = infv∈W s,p

0 (Ω)\0supt>0

Is,p(tv) > 0

The next lemma shows that the mountain pass levels of the ground state solutionsof (5.5) are uniformly bounded.

Lemma 5.9. Under the previous notation, cs0 ≥ lim sups→s0cs.

Proof. Let u ∈ Ns0,p, then tsuu ∈ Ns,p. We first need to analyze the behavior of tsu

as s → s0.Since tsuu ∈ Ns,p, we have that

〈(Js,p)′(tsuu), t

suu〉 =

n∑

i=1

|tsu|piJ i

si,pi(u)

=

∫

Ω

f(x, tsuu)tsuu dx

≥ µ

∫

Ω

F (x, tsuu) dx

= µ|tsu|pmax

∫

Ω

F (x, tsu)|u|pmax

|tsuu|pmax

dx.

Now, if we assume that tsu ≥ 1, we obtain that

pmaxJs,p(u) ≥ µ

∫

Ω

F (x, tsu)|u|pmax

|tsuu|pmax

dx

As, by Theorem 3.1, Js,p(u) → Js0,p(u) < ∞ when s → s0, from (f3) impliesthat tsus is bounded.


Let t0 ≥ 0 be any accumulation point of tsus. Let tk = tsku be a subsequencesuch that tk → t0 as k → ∞. Let us see that t0 6= 0. In fact, the Nehari identitygives us

n∑

i=1

|tk|piJ i

si,pi(u) =

∫

Ω

f(x, tku)tku dx

from where we get

n∑

i=1

|tk|pi−pminJ i

si,pi(u) =

∫

Ω

f(x, tku)

|tku|pmin−1|u|pmin−1u dx

Now, if t0 = 0 and pmin = pi0 , passing to the limit as k → ∞ using (f4) we deducethat

J i0si0 ,pmin

(u) = 0.

But this implies that u is independent of xi0 and thus that u = 0, a contradiction.Therefore t0 > 0.

Moreover, from (f2) and (f4) we have that

|f(x, tku)tku| ≤ ǫ|tku|pmin + Cǫ|tku|

q ≤ C(|u|pmin + |u|q) ∈ L1(Ω),

where C > 0 is independent of k. Then, using the dominated convergence theorem, ∫

Ω

f(x, tku)tku dx→

∫

Ω

f(x, t0u)t0u dx (as k → ∞).

So, we getn∑

i=1

|t0|piJ i

s0i,pi

(u) =

∫

Ω

f(x, t0u)t0u dx.

Since u ∈ Ns0,p, t0 = 1. Then tsu → 1 as s → s0.Now, we proceed as follows:

cs ≤ Is,p(tsuu) =

n∑

i=1

1

piJ isi,pi

(tsuu)−

∫

Ω

F (x, tsuu) dx

=

n∑

i=1

1

pi|tsu|

piJ isi,pi

(u)−

∫

Ω

F (x, tsuu) dx.

We can now take the limit as s → s0 in the former inequality to obtain

lim sups→s0

cs ≤n∑

i=1

1

piJ is0i,pi

(u)−

∫

Ω

F (x, u) dx = Is0,p(u),

where we have used Theorem 3.1.Then, we take the infimum in u ∈ Ns0,p to conclude the desired result.

Lemma 5.10. Let us ∈ W s,p0 (Ω) be a ground state solution of (5.5). Then there

exist two constants 0 < c < C <∞ independent on s such that

c ≤ Js,p(us) ≤ C


Proof. Let us ∈ W s,p0 (Ω) be a ground state solution of (5.5). By the previuos

lemma, there exist a constant C > 0, such that

C ≥ Is,p(us)−1

µ〈I ′s,p(us), us〉

= Js,p(us)−n∑

i=1

1

µJ isi,pi

(us)−

∫

Ω

(F (x, us)−

1

µf(x, us)us

)dx

≥n∑

i=1

(1

pi−

1

µ

)J isi,pi

(us)

≥

(1

pmax−

1

µ

)Js,p(u).

For the lower bound, since us ∈ Ns,p, we have that

(5.7) pminJs,p(us) ≤

∫

Ω

f(x, us)us dx ≤

∫

Ω

(ǫ|us|pmax + Cǫ|us|

q) dx.

Now from [7, Theorem 1.2], we obtain that

(5.8) min‖u‖pmin

p∗ , ‖u‖pmax

p∗ ≤ CJs,p(u),

where C is independent of u ∈W s,p0 (Ω).

Next, from (5.7), using Holder’s inequality and (5.8), we obtain

Js,p(us) ≤CǫmaxJs,p(us), Js,p(us)pmax/pmin

+ Cǫ maxJs,p(us)q/pmax , Js,p(us)

q/pmin.

Then, choosing ǫ > 0 small, we obtain that Js,p(us) is bounded below away fromzero, as we wanted to show.

Now we are ready to prove the main result of the section.

Theorem 5.11. Let sk → s0 as k → ∞ and let p be such that 1 < pmin ≤ pmax <p∗0 = p∗

s0. Assume that for each k ∈ N, there exists uk ∈ W sk,p

0 (Ω) a ground statesolution to (5.5).

Then the sequence ukk∈N is precompact in Lr(Ω), for every 1 < r < p∗0 and

any accumulation point u of the sequence ukk∈N belongs to W s0,p0 (Ω) and it is a

ground state solution to (5.6).

Proof. First observe that if sk → s0, then p∗k := p∗

sk→ p∗

0 as k → ∞. Then, ifr < p∗

0, then r < p∗k for every k large.

In view of (5.8) and Holder’s inequality, if we show that supk∈N Jsk,p(uk) <∞,then ukk∈N will be bounded in Lr(Ω).

But, this follows from Lemma 5.10 using that f satisfies (f6). Finally, Theorem3.3 gives the existence of a subsequence ukj

j∈N ⊂ ukk∈N and a function u ∈W s0,p

0 (Ω) such that ukj→ u0 in Lr(Ω).

It remains to see that u0 is a ground state solution to (5.6). But, by Theorem5.5, u0 is a weak solution of (5.6). Now, since uk ∈ Nsk,p, we have that

0 < c ≤ pminJsk,p(uk) ≤n∑

i=1

J isi,pi

(uk) =

∫

Ω

f(x, uk)uk dx→

∫

Ω

f(x, u0)u0 dx,


as k → ∞, so u0 6= 0 and it follows that u0 ∈ Ns0,p. Moreover,

lim infk→∞

csk = lim infk→∞

Isk,p(uk) = lim infk→∞

Jsk,p(uk)−

∫

Ω

F (x, uk)

≥ Js0,p(u0)−

∫

Ω

F (x, u0) dx = Is0,p(u0) ≥ cs0 ,

where we have used Corollary 3.5 in the inequality.Combining this estimate with Lemma 5.9 we conclude that

cs0 = Is0,p(u0)

and the proof is finished.

Acknowledgements

This work was partially supported by UBACYT Prog. 2018 20020170100445BA,by ANPCyT PICT 2019-00985 and by PIP No. 11220150100032CO.

J. Fernandez Bonder is a members of CONICET and I. Ceresa Dussel is a doc-toral fellow of CONICET.

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