Lorenzo Brasco
email: lorenzo.brasco@unife.it

Marco Squassina
email: marco.squassina@unicatt.it

OPTIMAL SOLVABILITY FOR A NONLOCAL PROBLEM AT CRITICAL GROWTH

Keywords: 2010 Mathematics Subject Classification. Primary 35R11, 35J92, 35B33, Secondary 35A15 Brezis-Nirenberg problem, critical growth problems, minimization problems

published or not. The documents may come

Optimal solvability for a nonlocal problem at critical growth

Lorenzo Brasco, Marco Squassina

1. Introduction 1.1. Overview. Let Ω be a bounded domain of R N with N ≥ 3. In 1983, in the celebrated paper [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], Brezis and Nirenberg studied the solvability conditions for the semi-linear elliptic problem (1.1)

     -∆u -λ u = u (N +2)/(N -2) , in Ω, u > 0,
in Ω, u = 0, on ∂Ω.

In particular, if λ 1 (Ω) denotes the first eigenvalue of the Dirichlet-Laplacian in Ω, they proved that, if N ≥ 4, then problem (1.1) admits a solution if 0 < λ < λ 1 (Ω) while for N = 3 there exists λ * ∈ (0, λ 1 ) such that (1.1) admits a solution if λ * < λ < λ 1 (Ω) and no solution for 0 < λ ≤ λ * . Due to this phenomenon, N = 3 is often referred to in the literature as critical dimension.

In general λ * is not given explicitly, except when Ω is a ball, in which case λ * = λ 1 (Ω)/4. In addition, there is no solution to (1.1) when λ ≥ λ 1 (Ω) for any domain Ω (see [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF]Remark 1.1]) and also for λ ≤ 0 provided Ω is smooth and star-shaped (see [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF]Remark 1.2]).

In the same paper the authors considered, for N ≥ 4, the non-autonomous critical elliptic problem (1.2)

     -∆u + a u = +u (N +2)/(N -2) , in Ω, u > 0,
in Ω, u = 0, on ∂Ω.

and obtained the existence of a solution by assuming that a ∈ L ∞ (Ω) and that there exist δ > 0 and an open subset Ω 0 ⊂ Ω such that

a ≤ -δ, in Ω 0 , ˆΩ |∇ϕ| 2 + a ϕ 2 dx ≥ δ ˆΩ ϕ 2 dx, for all ϕ ∈ C ∞ 0 (Ω),
see [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF]Section 4]. About the case of the critical dimension N = 3 for (1.2), no result is stated in [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF].

After the striking achievements of [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], many works were devoted to the search of solvability conditions for (possibly sign-changing) solutions of the problem -∆u -λ u = |u| 4/(N -2) u, in Ω, u = 0, on ∂Ω, Solution are obtained for any λ > 0 if N ≥ 5 and for all λ > 0 with λ ∈ σ(-∆) if N ≥ 4. Here σ(-∆) is the spectrum of the Dirichlet-Laplacian on Ω. We refer the reader e.g. to [7,12]. The main tool exploited is the Linking Theorem of Rabinowitz [18]. The previous results were then extended to the more general problem

(1.3) -∆ p u -λ |u| p-2 u = |u| p * -2 u in Ω, u = 0, on ∂Ω,
where 1 < p < N , -∆ p is the p-Laplacian operator and p * = N p/(N -p) is the critical Sobolev exponent. See for example [11,13] where positive solutions were found for N ≥ p 2 and 0 < λ < λ 1 (Ω), via the Mountain Pass theorem of Ambrosetti and Rabinowitz [18]. Here λ 1 (Ω) denotes the first eigenvalue of -∆ p with Dirichlet boundary conditions. For the general case of sign-changing solutions, a first result was obtained in [START_REF] Arioli | Some results on p-Laplace equations with a critical growth term[END_REF] for λ below the second eigenvalue λ 2 (Ω) of -∆ p under some restrictions on p and N . Recently, Degiovanni and Lancelotti in [START_REF] Degiovanni | Linking solutions for p-Laplace equations with nonlinearity at critical growth[END_REF] proved that, if the domain is of class C 1,α for some α ∈ (0, 1), then problem (1.3) admits a nontrivial solution for all λ > 0 provided that (N 3 + p 3 )/(N 2 + N ) > p 2 .

Recently, in the framework of nonlocal problems, the following Brezis-Nirenberg type problem for the fractional p-Laplacian was investigated

(1.4) (-∆ p ) s u -λ |u| p-2 u = |u| p * s -2 u, in Ω, u = 0, in R N \ Ω,
where s ∈ (0, 1), N > s p, λ > 0 and p * s = N p/(N -s p) is the fractional critical Sobolev exponent. In [START_REF] Mosconi | The Brezis-Nirenberg problem for the fractional p-Laplacian[END_REF] the authors proved, among other results, that problem (1.4) has a nontrivial weak solution for all λ > 0 provided that (N 3 + s 3 p 3 )/N (N + s) > s p 2 and Ω is of class C 1,1 .

Main results.

In this paper we return to the investigation of nonautonomous problems like (1.2), in the nonlinear nonlocal setting aiming to get optimal solvability conditions for the existence of ground state solutions. Let us set

[u] D s,p (R N ) := ˆR2N |u(x) -u(y)| p |x -y| N +s p dx dy 1/p
, and define the two spaces

D s,p (R N ) := {u ∈ L p * s (R N ) : [u] D s,p (R N ) < +∞}, D s,p 0 (Ω) := {u ∈ D s,p (R N ) : u = 0 on R N \ Ω}.
Precisely, for s p < N , we aim to study the solvability conditions for the following minimization problem

S p,s (a) := inf u∈D s,p 0 (Ω) [u] p D s,p (R N ) + ˆRN a |u| p dx : u L p * s (R N ) = 1 ,
where a ∈ L N/sp (Ω) is given. By Lagrange Multipliers Rule, minimizers of the previous problem (provided they exist) are constant sign weak solutions of

(1.5) (-∆ p ) s u + a |u| p-2 u = µ |u| p * s -2 u, in Ω, u = 0, in R N \ Ω,
with µ = S p,s (a). Namely, they satisfy

ˆR2N J p (u(x) -u(y)) ϕ(x) -ϕ(y) |x -y| N +s p dx dy + ˆRN a |u| p-2 u ϕ dx = µ ˆRN |u| p * s -2 u ϕ dx,
for every ϕ ∈ D s,p 0 (Ω). Throughout the paper we will use the notation for 1

< p < +∞ J p (t) = |t| p-2 t, t ∈ R.
We also introduce the sharp Sobolev constant

(1.6) S p,s := inf u∈D s,p (R N )\{0} [u] p D s,p (R N ) u p L p * s (R N )
.

Finally, we use the standard notations

a + = max{a, 0}, a -= max{-a, 0}, B R (x 0 ) = {x ∈ R N : |x -x 0 | < R}.
The main result of the paper is the following Theorem 1.1. Let Ω ⊂ R N be an open bounded set. The following facts hold:

1. If a ≥ 0, then S p,s (a) does not admit a solution.

2. Let N > s p 2 . Assume that there exist σ > 0, R > 0 and x 0 ∈ Ω such that

a -≥ σ, a. e. on B R (x 0 ) ⊂ Ω.
Then S p,s (a) admits a solution.

3. Let s p < N ≤ s p 2 . For any R > 0 there exists σ = σ(R, N, s, p) > 0 such that if

a -≥ σ, a. e. on B R (x 0 ) ⊂ Ω,
then S p,s (a) admits a solution.

The conditions on a for the existence of solutions in Theorem 1.1 are essentially optimal, see the discussion of Remark 4.1. In Proposition 3.4 we also prove that the solution is unique when S p,s (a) ≤ 0.

The precise form of the optimizers for the best constant in the fractional Sobolev embedding is still unknown (and proving it seems currently out of reach), although minimizers were conjectured to be of the form

c U x -x 0 ε , where U (x) = (1 + |x| p ) -(N -sp)/p , x ∈ R N , c = 0, x 0 ∈ R N and ε > 0.
In the proof of Theorem 1.1, a key tool is the use of suitable truncations of extremals U of the Sobolev inequality, introduced in [START_REF] Mosconi | The Brezis-Nirenberg problem for the fractional p-Laplacian[END_REF] (see Section 2 below). In fact, the knowledge of their decay at infinity (recently proved in [START_REF] Brasco | Optimal decay of extremals for the fractional Sobolev inequality[END_REF]) is enough to conclude.

Then, we also have the following result Theorem 1.2. Let Ω ⊂ R N be an open bounded set. Let a ∈ L N/sp (Ω) be such that

(1.7) S p,s (a) < 0.
Then problem (1.5) i) does not admit positive solutions if µ ≥ 0;

ii) admits a unique positive solution if µ < 0.

Remark 1.3. We can rephrase condition (1.7) also in terms of the following Poincaré-type constant

λ(Ω, a) := inf u∈D s,p 0 (Ω) [u] p D s,p (R N ) + ˆRN a + |u| p dx : ˆRN a -|u| p dx = 1 .
Indeed, we can show that λ(Ω; a) < 1 ⇐⇒ S p,s (a) < 0, and also λ(Ω; a) = 1 ⇐⇒ S p,s (a) = 0, see Remark 3.5. For a discussion on sufficient conditions ensuring λ(Ω; a) < 1, and hence (1.7), we refer the reader to Remark 5.1.

Remark 1.4 (The case of the p-Laplacian). Although we can only formally choose the limiting value s = 1 in the previous statements, the same proofs in the paper would provide existence and non-existence results for the following quasilinear local problem

S p (a) = inf u∈D 1,p 0 (Ω) ˆRN |∇u| p dx + ˆRN a |u| p dx : u L N p/(N -p) (R N ) = 1 .
Finally, we stress that already in the semi-linear case p = 2, Theorem 1.1 and Theorem 1.2 cover situations which were previously open, such as the critical case of dimension N = 3 and also the case of weaker integrability assumptions on the external potential a.

Preliminaries

2.1. Some known results. We start with an elementary inequality, whose proof is based on Calculus and we omit it.

Lemma 2.1. If γ < 0, then we have

(2.1) (1 -t) γ ≤ 1 + 2 t (2 -γ -1), for every 0 ≤ t ≤ 1 2 .
If 0 ≤ γ ≤ 1, then we have

(2.2) (1 -t) γ ≥ 1 + 2 t (2 -γ -1), for every 0 ≤ t ≤ 1 2 .
The following is a discrete version of the celebrated Picone's inequality. See [2, Proposition 4.2] and [START_REF] Frank | Non-linear ground state representations and sharp Hardy inequalities[END_REF]Lemma 2.6] for a proof. We recall that in [3, Theorem 1.1] the following result was proved.

Theorem 2.3 (Decay of extremals). Let U ∈ D s,p (R N ) be any minimizer for (1.6). Then U ∈ L ∞ (R N ) is a constant sign, radially symmetric and monotone function with

lim |x|→∞ |x| N -s p p-1 U (x) = U ∞ , for some constant U ∞ ∈ R \ {0}.
Let us now fix U ∈ D s,p (R N ) a positive minimizer of (1.6), such that

[U ] p D s,p (R N ) = U p * s L p * s (R N ) = S N s p p,s , and
U (0) = 1.
Since this is a radially symmetric function, with a slight abuse of notation we write U (r) in place of U (x) with |x| = r. Based upon the above decay estimate, we can infer the following (see [START_REF] Mosconi | The Brezis-Nirenberg problem for the fractional p-Laplacian[END_REF]Lemma 2.2]).

Lemma 2.4. With the notation above, there exists θ > 1 such that for all r ≥ 1,

U (θ r) U (r) ≤ 1 2 .
In Section 3 we will need to use some truncations of the Sobolev extremal U . To this aim, for ε > 0 we first set

U ε (r) := ε s p-N p U r ε , r ≥ 0.
which still solves (1.6). Then by following [START_REF] Mosconi | The Brezis-Nirenberg problem for the fractional p-Laplacian[END_REF], for δ > 0 we introduce

(2.4) u ε,δ (r) :=        U ε (r), r ≤ δ U ε (δ) U ε (r) -U ε (θδ) U ε (δ) -U ε (θ δ) , δ < r ≤ θ δ 0, r > θ δ,
where θ is the constant appearing in Lemma 2.4. We then recall from [16, Lemma 2.7] the following crucial energy estimates for u ε,δ .

Lemma 2.5 (Energy estimates).

There exists C = C(N, p, s) > 0 such that for any ε ≤ δ/2,

[u ε,δ ] p D s,p (R N ) ≤ (S p,s ) N sp + C ε δ N -s p p-1 , u ε,δ p L p * s (R N ) ≥ (S p,s ) N sp -C ε δ N/(p-1) N -s p N .
2.2. Levels of compactness. The next result tells us that, in order to have loss of compactness, a certain amount of energy is necessary.

Lemma 2.6. Let Ω ⊂ R N be an open set and let us suppose that s p < N . Let µ ∈ R \ {0}, the functional

(2.5) K(u) := 1 p [u] p D s,p (R N ) + 1 p ˆΩ a |u| p dx - µ p * s ˆΩ |u| p * s dx, u ∈ D s,p 0 (Ω),
satisfies the Palais-Smale condition:

• at every energy level c ∈ R, if µ < 0;

• at every energy level c such that

c < s N µ S p,s µ N s p , if µ > 0.
Proof. We discuss the two cases separately.

Case µ < 0. Let {u n } n∈N ⊂ D s,p 0 (R N ) be a Palais-Smale sequence at level c, i.e. 1 p [u n ] p D s,p (R N ) + 1 p ˆΩ a |u n | p dx - µ p * s ˆΩ |u n | p * s dx = c + o n (1)
and sup

ϕ D s,p 0 (R N ) =1 ˆR2N J p (u n (x) -u n (y)) ϕ(x) -ϕ(y) |x -y| N +s p dx dy + ˆΩ a |u n | p-2 u n ϕ dx -µ ˆΩ |u n | p * s -2 u n ϕ dx = o n (1).
(2.6)

The first condition implies that for n sufficiently large we have

c + 1 ≥ 1 p [u n ] p D s,p (R N ) - 1 p a -L N/sp (Ω) ˆΩ |u n | p * s dx p p * s - µ p * s ˆΩ |u n | p * s dx ≥ 1 p [u n ] p D s,p (R N ) - p * s -p p p * s ε -p p * s -p a - p * s p * s -p L N/sp (Ω) - µ + ε p * s ˆΩ |u n | p * s dx,
where we used Hölder's and Young's inequalities. By choosing ε = -µ > 0, we get that the sequence is bounded in D s,p 0 (Ω). Thus we can infer weak convergence in D s,p 0 (Ω) and L p * s (Ω) (up to a subsequence) to a function u ∈ D s,p 0 (Ω). By weak convergence, we obtain

ˆR2N J p (u(x) -u(y)) (u n -u)(x) -(u n -u)(y) |x -y| N +s p dx dy + ˆRN a |u| p-2 u (u n -u) dx -µ ˆRN |u| p * s -2 u (u n -u) dx = o n (1).
From (2.6) and using that {u n } n∈N is bounded in D s,p 0 (Ω), we obtain

ˆR2N J p (u n (x) -u n (y)) (u n -u)(x) -(u n -u)(y) |x -y| N +s p dx dy + ˆRN a |u n | p-2 u n (u n -u) dx -µ ˆRN |u n | p * s -2 u n (u n -u) dx = o n (1).
By subtracting the last two displays, we obtain

ˆR2N J p (u n (x) -u n (y)) -J p (u(x) -u(y)) (u n -u)(x) -(u n -u)(y) |x -y| N +s p dx dy + ˆRN a |u n | p-2 u n -|u| p-2 u (u n -u) dx -µ ˆRN |u n | p * s -2 u n -|u| p * s -2 u (u n -u) dx = o n (1)
.

By using that -µ > 0 and the monotonicity of t → |t| p * s -2 t, we get

ˆR2N J p (u n (x) -u n (y)) -J p (u(x) -u(y)) (u n -u)(x) -(u n -u)(y) |x -y| N +s p dx dy + ˆRN a |u n | p-2 u n -|u| p-2 u (u n -u) dx ≤ o n (1).
(2.7)

We now observe that by Lemma 2.8 below

ˆRN a |u n | p-2 u n -|u| p-2 u (u n -u) dx = o n (1),
and that

J p (u n (x) -u n (y)) -J p (u(x) -u(y)) (u n -u)(x) -(u n -u)(y) ≥ 0,
by monotonicity of t → J p (t). Thus from (2.7) we can finally infer

ˆR2N J p (u n (x) -u n (y)) -J p (u(x) -u(y)) (u n -u)(x) -(u n -u)(y) |x -y| N +s p dx dy = o n (1)
.

By standard monotonicity inequalities, we eventually infer the strong convergence to u.

Case µ > 0. It is an easy variant of the proof of [START_REF] Perera | Bifurcation and multiplicity results for critical fractional p-Laplacian problems[END_REF]Proposition 3.1] where the compactness is obtained for a constant potential a ≡ -µ, where µ > 0.

Remark 2.7. When µ = 0, in general the functional

K(u) := 1 p [u] p D s,p (R N ) + 1 p ˆRN a |u| p dx, u ∈ D s,p 0 (Ω),
does not satisfy the Palais-Smale condition, unless some conditions on a are imposed. For example, let us define the first eigenvalue of the fractional p-Laplacian, i.e.

(2.8)

λ 1 (Ω) := inf u∈D s,p 0 (Ω) [u] p D s,p (R N ) : ˆRN |u| p dx = 1 .
Then by taking a = -λ 1 (Ω), and φ 1 a minimizer of (2.8), the sequence

u n = n φ 1 , n ∈ N,
is a Palais-Smale sequence for K (at level 0), but the sequence is not even bounded.

It is easily seen that in this case K satisifes the Palais-Smale condition at every level, when

a -L N/sp (Ω) < S p,s . Lemma 2.8. Let 1 < p < ∞ and s ∈ (0, 1) be such that s p < N . Let Ω ⊂ R N be an open bounded set. Let {u n } ⊂ D s,p 0 (Ω) be such that [u n ] D s,p (R N ) ≤ C, for every n ∈ N.
Then there exists u ∈ D s,p 0 (Ω) such that (up to a subsequence)

|u n | p-2 u n -|u| p-2 u (u n -u) n∈N , converges weakly to 0 in L p * s /p (Ω).
Proof. The hypothesis implies there exists u ∈ D s,p 0 (Ω) such that (up to a subsequence) the sequence converges weakly in D s,p 0 (Ω) and strongly in L p (Ω) to u, by compactness of the embedding D s,p 0 (Ω) → L p (Ω). In particular, up to extracting a further subsequence, we can suppose that u n converges almost everywhere to u.

We now observe that

|u n | p-2 u n -|u| p-2 u (u n -u) ≤ C    |u n -u| p , if 1 < p ≤ 2, |u n | p-2 + |u| p-2 |u n -u| 2 , if p > 2,
for some C = C(p) > 0. By using Sobolev and Hölder inequalitites, this implies that the sequence

v n := |u n | p-2 u n -|u| p-2 u (u n -u),
is bounded in L p * s /p (Ω). Moreover, v n converges almost everywhere to 0 by the first part of the proof. Then the conclusion follows from [14, Lemme 4.8].

Analysis of the ground state level

Throughout this section, Ω will be an open bounded set and we will assume s p < N . For a given potential a ∈ L N/sp (Ω), we want to study some properties of the ground state level S p,s (a) := inf

u∈D s,p 0 (Ω) [u] p D s,p (R N ) + ˆRN a |u| p dx : u L p * s (R N ) = 1 .
We first observe that S p,s (a) > -∞, indeed for every admissible function u we have

[u] p D s,p (R N ) + ˆRN a |u| p dx ≥ [u] p D s,p (R N ) -a L N/sp (Ω) ≥ S p,s -a L N/sp (Ω) ,
by Hölder's and Sobolev inequalities. We observe that when a ≡ 0, the problem above coincides with the determination of the best constant in the inequality

[u] p D s,p (R N ) ≥ c u p L p * s (Ω) , for u ∈ D s,p 0 (R N ).
As in the local case, such a constant does not depend on Ω and is never attained on proper subsets of R N . This is the content of the next result.

(Ω), whenever |R N \ Ω| > 0. Proof. Let ε > 0, then there exists ϕ ε ∈ C ∞ 0 (R N ) such that ϕ ε L p * s (R N ) = 1 and [ϕ ε ] D s,p (R N ) ≤ S p,s + ε. Let λ > 1, by taking ϕ λ,ε (x) = λ N -s p p ϕ ε (λ x),
we still have that

ϕ λ,ε L p * s (R N ) = 1 and [ϕ λ,ε ] D s,p (R N ) ≤ S p,s + ε.
Moreover, by taking λ large enough and up to translations, we have ϕ λ,ε ∈ C ∞ 0 (Ω). Thus we can use it as a test function in the problem defining S p,s (0) and obtain

S p,s (0) ≤ [ϕ λ,ε ] D s,p (R N ) ≤ S p,s + ε.
By arbitrariness of ε, this gives S p,s (0) ≤ S p,s . The reverse inequality is straightforward, thanks to the embedding D s,p 0 (Ω) → D s,p (R N ). We now suppose that S p,s (0) is attained by u ∈ D s,p 0 (Ω) \ {0}. By extending it to 0 outside Ω, we have u ∈ D s,p (R N ) and thus u is an extremal for the Sobolev inequality on the whole R N , thanks to the first part of the proof. Observe that u can be taken to be non-negative, due to Proposition 3.2 below. By optimality, we get that u is a constant sign supersolution of (-∆ p ) s , i.e. (-∆ p ) s u ≥ 0, in R N . Thus by the minimum principle of [2, Proposition A.1], we should have u > 0 almost everywhere. But this contradicts the fact that u ≡ 0 in R N \ Ω. Proposition 3.2. Let us assume that the variational problem defining S p,s (a) admits a solution u ∈ D s,p 0 (Ω). Then u has constant sign and u = 0 almost everywhere in Ω. Proof. We observe that the function |u| is still admissible for the variational problem and

|u| D s,p (R N ) ≤ [u] D s,p (R N ) ,
with inequality being strict if both u + and u -are nontrivial. By minimality of u, this implies that u must have constant sign. Let us assume for example that u ≥ 0 almost everywhere in Ω, then by a simple application of the Lagrange Multipliers Rule we get that u is a non negative solution of (1.5), where µ = S p,s (a). By appealing to the minimum principle of [START_REF] Brasco | Global compactness results for nonlocal problems[END_REF]Proposition B.3] we get that u > 0 almost everywhere.

Proposition 3.3. Let µ ≤ 0, then the problem      (-∆ p ) s u + a u p-1 = µ u p * s -1 , in Ω, u > 0, in Ω, u = 0, in R N \ Ω,
admits at most • one solution, if µ < 0;

• one solution with unit L p * s norm, if µ = 0.

Proof. We use an idea due to Brezis and Oswald, based on Picone's inequality (see [START_REF] Brezis | Remarks on sublinear elliptic equations[END_REF]). We assume that the problem above admits two solutions u 1 , u 2 ∈ D s,p 0 (Ω). We fix ε > 0 and define u i,ε = min{u i , 1/ε} for i = 1, 2. We have

ˆR2N J p (u 1 (x) -u 1 (y)) ϕ(x) -ϕ(y) |x -y| N +s p dx dy + ˆRN a u p-1 1 ϕ dx = µ ˆRN u p * s -1 1 ϕ dx, ˆR2N J p (u 2 (x) -u 2 (y)) ϕ(x) -ϕ(y) |x -y| N +s p dx dy + ˆRN a u p-1 2 ϕ dx = µ ˆRN u p * s -1 2 ϕ dx.
We test these equations respectively with

ϕ 1 := u p 2,ε (u 1 + ε) p-1 -u 1,ε , ϕ 2 := u p 1,ε (u 2 + ε) p-1 -u 2,ε .
By adding the two resulting identities, we obtain

ˆR2N J p (u 1 (x) -u 1 (y)) u p 2,ε (u 1 + ε) p-1 (x) - u p 2,ε (u 1 + ε) p-1 (y) |x -y| N +s p dx dy - ˆR2N J p (u 1 (x) -u 1 (y)) u 1,ε (x) -u 1,ε (y) |x -y| N +s p dx dy + ˆR2N J p (u 2 (x) -u 2 (y)) u p 1,ε (u 2 + ε) p-1 (x) - u p 1,ε (u 2 + ε) p-1 (y) |x -y| N +s p dx dy - ˆR2N J p (u 2 (x) -u 2 (y)) u 2,ε (x) -u 2,ε (y) |x -y| N +s p dx dy + ˆRN a u p-1 1 u p 2,ε (u 1 + ε) p-1 -u 1,ε dx + ˆRN a u p-1 2 u p 1,ε (u 2 + ε) p-1 -u 2,ε dx = µ ˆRN u p * s -1 1 u p 2,ε (u 1 + ε) p-1 -u 1,ε dx + µ ˆRN u p * s -1 2 u p 1,ε (u 2 + ε) p-1 -u 2,ε dx. (3.1)
We now observe that

J p (u i (x) -u i (y)) = J p ((u i + ε)(x) -(u i + ε)(y)), for i = 1, 2,
thus by Picone's inequality Proposition 2.2 we have

J p (u 1 (x) -u 1 (y)) u p 2,ε (u 1 + ε) p-1 (x) - u p 2,ε (u 1 + ε) p-1 (y) ≤ |u 2 (x) -u 2 (y)| p ,
where we also used that t → min{|t|, 1/ε} is 1-Lipschitz. Similarly, we get

J p (u 2 (x) -u 2 (y)) u p 1,ε (u 2 + ε) p-1 (x) - u p 1,ε (u 2 + ε) p-1 (y) ≤ |u 1 (x) -u 1 (y)| p .
We then pass to the limit in (3.1), by using Fatou's Lemma in the first and third terms and the Dominated Convergence Theorem in all the others. This yields

ˆR2N J p (u 1 (x) -u 1 (y)) u p 2 u p-1 1 (x) - u p 2 u p-1 1 (y) |x -y| N +s p dx dy - ˆR2N |u 1 (x) -u 1 (y)| p |x -y| N +s p dx dy + ˆR2N J p (u 2 (x) -u 2 (y)) u p 1 u p-1 2 (x) - u p 1 u p-1 2 (y) |x -y| N +s p dx dy - ˆR2N |u 2 (x) -u 2 (y)| p |x -y| N +s p dx dy ≥ µ ˆRN u p * s -p 1 u p 2 dx -µ ˆRN u p * s 1 dx + µ ˆRN u p * s -p 2 u p 1 dx -µ ˆRN u p * s 2 dx. (3.2)
We now use Picone's inequality in the left-hand side of (3.2). This gives 0 ≥ -µ

ˆRN (u p 1 -u p 2 ) (u p * s -p 1 -u p * s -p 2 ) dx.
If µ < 0, from the previous inequality we directly get that u 1 = u 2 .

If µ = 0, we go back to (3.2) and observe that this and Picone's inequality imply

ˆR2N J p (u 1 (x) -u 1 (y)) u p 2 u p-1 1 (x) - u p 2 u p-1 1 (y) |x -y| N +s p dx dy = ˆR2N |u 1 (x) -u 1 (y)| p |x -y| N +s p dx dy.
By appealing to equality cases in Picone's inequality, we get the conclusion

u 1 (x) u 1 (y) = u 2 (x) u 2 (y)
, for a. e. x, y ∈ Ω.

This implies that u 1 = c u 2 for some positive constant c.

In the local case, formally corresponding to s = 1, the assertion below was proved in [START_REF] Mercuri | A global compactness result for the p-Laplacian involving critical nonlinearities[END_REF].

Proposition 3.4. Let us suppose that S p,s (a) < S p,s . Then the problem defining S p,s (a) has a solution. Moreover, if S p,s (a) ≤ 0, then such a solution is unique, up to the choice of the sign.

Proof. We first observe that if S p,s (a) ≤ 0, then the uniqueness follows by combining Propositions 3.2 and 3.3, since every minimizer is a constant sign solution of (1.5), with µ = S p,s (a).

We now come to the existence part and divide the proof in two cases.

Case S p,s (a) = 0. Let {u n } n∈N ⊂ D s,p 0 (Ω) be a sequence such that By Lemma 2.6, we can infer strong convergence of the minimizing sequence { u n } n∈N in D s,p 0 (Ω) and thus existence of a solution u to the problem defining S p,s (a).

u n L p * s (R N ) = 1 and lim n→∞ [u n ] p D s,p (R N ) + ˆRN a |u n | p dx = S p,s (a)
J( u n ) ≤ J(u n ) and G( u n ) = G(u n ) = 1 p * s , and
lim n→∞   sup ϕ D s,p 0 (R N ) =1 J ( u n ) -λ n G ( u n ), ϕ   = 0. Since the sequence { u n } n∈N is bounded in D s,p 0 (Ω), we have J ( u n ), u n -λ n G ( u n ), u n = o n (
Case S p,s (a) = 0. We first observe that if a ≥ 0 on Ω, then S p,s (a) ≥ S p,s > 0.

Thus S p,s (a) = 0 implies that a -≡ 0. By definition of S p,s (a) and the fact that S p,s (a) = 0, this implies that we have the Poincaré-type inequality

(3.3) [u] p D s,p (R N ) + ˆRN a + |u| p dx ≥ ˆRN a -|u| p dx,
for every u ∈ D s,p 0 (Ω). Let us consider the sharp constant in the previous Poincaré-type inequality, i.e. (3.4) λ(Ω, a)

:= inf [u] p D s,p (R N ) + ˆRN a + |u| p dx : ˆRN a -|u| p dx = 1 .
It is standard routine to see that the infimum above is achieved, by a constant-sign function. Let us call φ 1 the positive solution. Observe that by (3.3), we have λ(Ω, a) ≥ 1. On the other hand, since S p,s (a) = 0 there exists a sequence {u n } n∈N ⊂ D s,p 0 (Ω) such that

(3.5) u n L p * s (R N ) = 1 and [u n ] p D s,p (R N ) + ˆRN a + |u n | p dx - ˆRN a -|u n | p dx = o n (1)
.

We now observe that by Sobolev inequality we have

ˆRN a -|u n | p dx = [u n ] p D s,p (R N ) + ˆRN a + |u n | p dx + o n (1) ≥ S p,s + o n (1),
where we used that every u n has unit norm in L p * s . We can thus divide the second equation in (3.5) by ´RN a -|u n | p dx and obtain

λ(Ω, a) ≤ lim n→∞ [u n ] p D s,p (R N ) + ˆRN a + |u n | p dx ˆRN a -|u n | p dx = 1.
This finally implies that λ(Ω, a) = 1 and thus the function

v 0 := φ 1 φ 1 L p * s (R N ) , is such that [v 0 ] p D s,p (R N ) + ˆRN a |v 0 | p-2 v 0 dx = 0 and v 0 L p * s (R N ) = 1,
i.e. v 0 is a solution of the problem defining S p,s (a).

Remark 3.5. The quantity λ(Ω, a) defined by (3.4) is the first eigenvalue of the following eigenvalue problem

(-∆ p ) s u + a + |u| p-2 u = λ a -|u| p-2 u, in Ω, u = 0, in R N \ Ω.
It is not difficult to see that

S p,s (a) = 0 ⇐⇒ λ(Ω, a) = 1.
Indeed, the implication =⇒ has been proven above. The converse implication goes as follows: we suppose λ(Ω, a) = 1 and take φ 1 a solution of problem (3.4). From inequality (3.3), we have

[u] p D s,p (R N ) + ˆRN a |u| p dx ≥ 0,
for every u ∈ D s,p 0 (Ω) with unit L p * s norm. This implies that S p,s (a) ≥ 0. On the other hand, the function φ 1 gives equality in the previous inequality, thus

S p,s (a) ≤ [φ 1 ] p D s,p (R N ) + ˆRN a |φ 1 | p dx ˆRN |φ 1 | p * s dx p p * s = 0,
as well. Actually, we can also show that

(3.6) S p,s (a) < 0 ⇐⇒ λ(Ω; a) < 1.
If S p,s (a) < 0, by the previous result there exists a minimizer φ for the problem defining S p,s (a).

Thus in particular we get

[φ] p D s,p (R N ) + ˆRN a + |φ| p dx - ˆRN a -|φ| p dx = S p,s (a) < 0,
which immediately implies λ(Ω; a) < 1. On the other hand, the minimizer φ 1 of problem (3.4) now verifies

[φ 1 ] p D s,p (R N ) + ˆRN a + |φ 1 | p dx = λ(Ω; a) ˆRN a -|φ 1 | p dx < ˆRN a -|φ 1 | p dx.
By using the function φ 1 / φ 1 L p * s as a competitor for the problem defining S p,s (a) and appealing to the previous estimate, we then get S p,s (a) < 0.

Remark 3.6. It is not difficult to see that

a -L N/sp (Ω) < S p,s =⇒ S p,s (a) > 0.
Indeed, by Hölder's and Sobolev inequalities

[u] p D s,p (R N ) + ˆRN a |u| p dx ≥ [u] p D s,p (R N ) - ˆRN a -|u| p dx ≥ S p,s u p L p * s -a -L N/sp (Ω) u p L p * s = S p,s -a -L N/sp (Ω) > 0.
for every function with unit L p * s norm.

Proof of Theorem 1.1

We proceed to prove each point separately.

1. Case a ≥ 0. Let us prove that for a ≥ 0 the problem does not admit any solution. By Lemma 3.1, we already know that this is true if a ≡ 0, thus let us assume that a is non-negative and

a L N/sp (Ω) > 0.
It is sufficient to show that in this case S p,s (a) ≤ S p,s . Indeed, let us assume the latter to be true. If u ∈ D s,p 0 (Ω) is a solution for the problem defining S p,s (a), we would get

S p,s ≥ S p,s (a) = [u] p D s,p (R N ) + ˆRN a |u| p dx ≥ [u] p D s,p (R N ) ≥ S p,s .
Then we have equalities everywhere and in particular ˆRN a |u| p dx = 0, which implies that u = 0 almost everywhere on the support of a. This contradicts the properties of u contained in Proposition 3.2.

In order to prove S p,s (a) ≤ S p,s , we consider the functions u ε,δ ∈ D s,p 0 (R N ) as defined in (2.4). We fix δ > 0 small enough so that, up to translations, u ε,δ has support contained in Ω. Then accordingly we take ε ≤ δ/2. We have

(4.1) S p,s (a) ≤ [u ε,δ ] p D s,p (R N ) + ˆRN a |u ε,δ | p dx u ε,δ p L p * s (R N )
.

By the estimates of Lemma 2.5 and

1 lim ε→0 ˆRN a |u ε,δ | p dx = 0,
when we let ε go to 0 in (4.1), we get S p,s (a) ≤ S p,s .

2. Case N > s p 2 . We want to prove that in this case S p,s (a) < S p,s and then apply Proposition 3.4. By assumption, there exist σ > 0, R > 0 and x 0 ∈ Ω such that

a(x) ≤ -σ for a. e. x ∈ B R (x 0 ) ⊂ Ω.
Without loss of generality, we can assume that x 0 = 0. Let θ > 1 be the same constant appearing in Lemma 2.3, we choose δ = R/θ. Then we consider again the functions u ε,δ ∈ D s,p 0 (Ω) as defined in (2.4), with

(4.2) ε ≤ ε 0 := δ min    1 2 , 1 2
(S p,s ) N/sp C p-1 N    ,
that will be chosen in a while. The constant C is the same as in Lemma 2.5. We fix α such that

s p < α < N -s p p -1 ,
a choice which is feasible thanks to the assumption N > s p 2 . We now set

a ε (x) := -ε α u p * s -p ε,δ (x) ∈ L N/sp (Ω),
and we enforce condition (4.2), by requiring that ε > 0 satisfies

ε ≤ ε 1 := min ε 0 , σ 1 α-s p .
Observe that with such a choice, we have (recall that we are taking U (0) = 1)

a ε (x) = -ε α u p * s -p ε,δ (x) ≥ -ε α u p * s -p ε,δ (0) = -ε α U p * s -p ε (0) ≥ -σ, in B R (0).
Thanks to the hypothesis on a, this yields a(x) ≤ a ε (x), for a. e. x ∈ B R (0).

Then by using the test function u ε,δ (which is supported in the ball B R (0) ⊂ Ω) we get

S p,s (a) ≤ [u ε,δ ] p D s,p (R N ) + ˆRN a |u ε,δ | p dx u ε,δ p L p * s (R N ) = [u ε,δ ] p D s,p (R N ) + ˆBR (0) a |u ε,δ | p dx u ε,δ p L p * s (R N ) ≤ [u ε,δ ] p D s,p (R N ) + ˆBR (0) a ε |u ε,δ | p dx u ε,δ p L p * s (R N ) =: R(ε, δ).
We need to estimate the last Rayleight-type quotient. Thanks to the definition of a ε and Lemma 2.5 we can infer

R(ε, δ) = [u ε,δ ] p D s,p (R N ) u ε,δ p L p * s (R N ) -ε α u ε,δ p * s -p L p * s (R N ) ≤ (S p,s ) N sp + C ε δ N -s p p-1 (S p,s ) N sp -C ε δ N p-1 N -s p N -ε α (S p,s ) N sp -C ε δ N p-1 s p N .
We now apply (2.1) with

γ = - N -s p N , t = C (S p,s ) N sp ε δ N p-1 and (2.2) with γ = s p N , t = C (S p,s ) N sp ε δ N p-1 .
This gives

R(ε, δ) ≤ S p,s 1 + C (S p,s ) N sp ε δ N -s p p-1 1 + 2 C (S p,s ) N sp ε δ N p-1 (2 N -s p N -1) -ε α S p,s 1 + 2 C (S p,s ) N sp ε δ N p-1 (2 -s p N -1) = S p,s + ε α -S p,s + C (S p,s ) N sp -1 ε δ N -s p p-1 -α + 2 C (S p,s ) N sp -1 (2 N -s p N -1) ε δ N p-1 -α - 2 C (S p,s ) N sp -1 ε δ N p-1 (2 -s p N -1) + 2 S p,s C (S p,s ) N sp 2 ε δ 2 N -s p p-1 -α   .
By recalling the choice of α, we have that there exists ε 2 = ε 2 (N, s, p, δ) > 0 such that for every 0 < ε ≤ ε 2 the term above into square brackets is negative 2 . Thus in particular S p,s (a) ≤ R(ε, δ) < S p,s , for every 0 < ε ≤ min{ε 1 , ε 2 }.

Then the existence of a solution follows from Proposition 3.4.

3.

Case s p < N ≤ s p 2 . Let us prove the second assertion. We fix x 0 ∈ Ω and R > 0 such that B R (x 0 ) ⊂ Ω. As before, we assume for simplicity that x 0 = 0. Then we choose 0 ≤ α < N -s p p -1 .

Observe that now we automatically get α < s p as well. We set again δ = R/θ and

a ε (x) := -ε α u p * s -p ε,δ (x)
. By arguing as in the estimate above for the term R(ε, δ), we gain the existence of an explicit

ε 2 > 0 such that [u ε 2 ,δ ] p D s,p (R N ) + ˆRN a ε 2 |u ε 2 ,δ | p dx u ε 2 ,δ p L p * s (R N ) < S p,s .
Observe that ε 2 can be taken so that ε 2 ≤ δ/2. By construction, we have

a ε 2 L ∞ (B δ (0)) = U (0) p * s -p ε α-s p 2 = ε α-s p 2 ,
and observe that α -s p < 0. If we assume that

a -≥ σ := ε α-s p 2 , a. e. in B R (0),
we have a ≤ -σ ≤ a ε 2 a. e. in B R (0). We use u ε 2 ,δ as a test function. By using the previous estimate and the fact that u ε 2 ,δ is supported on B R (0), we get as before S p,s (a) < S p,s . By appealing again to Proposition 3.4, we can infer that S p,s (a) has a solution.

Remark 4.1 (About the condition on a). In order to understand optimality of the conditions on a, let us consider the case s = 1, p = 2 and a = -λ for some constant λ > 0. In this case, solutions φ of

(4.3) S 2,1 (-λ) := inf u∈D 1,2 0 (Ω) ˆΩ |∇u| 2 dx -λ ˆΩ |u| 2 dx : u L 2 N N -2 (Ω) = 1 , verifies in weak sense -∆φ -λ φ = µ φ N +2 N -2
in Ω, φ > 0 in Ω, φ = 0 on ∂Ω, for µ = S 2,1 (-λ). Observe that the sign of the Lagrange multiplier µ depends on whether λ < λ 1 (Ω) (in this case µ > 0) or λ > λ 1 (Ω) (in this case µ < 0). Here λ 1 (Ω) is the first eigenvalue of the Dirichlet-Laplacian on Ω. In the first case, by setting

ψ = µ N -2 4
φ, 2 The constant ε2 is easily seen to have the following form

ε2 = Sp,s C 1 N -s p p-1 -α R θ ,
with C = C(N, p, s) > 0 that can be computed explicitely.

we would get a nontrivial solution of

-∆ψ -λ ψ = ψ 2 N N -2 -1 in Ω, ψ ≥ 0 in Ω, ψ = 0 on ∂Ω.
We already recalled in the Introduction that a necessary condition for this to be possible is that

λ > λ * > 0 if N = 3, λ > 0 if N ≥ 4,
for a suitable λ * < λ 1 (Ω). In other words, by observing that in this case we (formally) have s p = 2 and s p 2 = 4, problem (4.3) can not have a solution for s p < N < s p 2 if λ > 0 is arbitrarily small, while for N ≥ s p 2 the parameter λ > 0 can be taken as small as desired. This exactly fits into the statement of Theorem 1.1 when a is a negative constant.

For completeness, we also record the following results. Proof. We pick u ∈ D s,p 0 (Ω) with u L p * s (Ω) = 1. Then, Hölder's inequality yields

S p,s (a) ≤ [u] p D s,p (R N ) + ˆRN a |u| p dx ≤ [u] p D s,p (R N ) + ˆRN a |u| p dx + a -a L N/sp (Ω) .
Thus S p,s (a) ≤ S p,s (a ) + a -a L N/sp (Ω) . Switching the role of a and a , we get (4.4).

[u k ] p D s,p (R N ) + ˆRN a k |u k | p dx = S p,s (a),
thus, without loss of generality, we can assume

[u k ] p D s,p (R N ) + ˆRN a k |u k | p dx ≤ S p,s (a) + 1.
Moreover, by hypothesis we have a k L N/sp (Ω) ≤ M, thus by Hölder's inequality we get

S p,s (a) + 1 ≥ [u k ] p D s,p (R N ) + ˆRN a k |u k | p dx ≥ [u k ] p D s,p (R N ) -M.
This implies that the sequence {u k } k∈N is equi-bounded in D s,p 0 (Ω) and thus we can infer strong convergence (up to a subsequence) in L q (Ω), for q < p * s , to some limit function u ∈ D s,p 0 (Ω). We need to show that (4.6)

u L p * s (R N ) = 1 and S p,s (a) ≥ [u] p D s,p (R N ) + ˆRN a |u| p dx.
The second fact easily follows from (4.5), the lower semicontinuity of the Gagliardo seminorm and the weak convergence of {|u k | p } k∈N in L p * s /p (Ω) to |u| p . Observe that the conditions (4.6) automatically give that u is a minimizer for S p,s (a).

In order to conclude, we need to prove that {u k } k∈N converges strongly in L p * s (Ω). Observe that by minimality of u k , we get

ˆR2N |u k (x) -u k (y)| p-2 (u k (x) -u k (y)) (ϕ(x) -ϕ(y)) |x -y| N +s p dx dy + ˆRN a k |u k | p-2 u k ϕ dx = S p,s (a k ) ˆRN |u k | p * s -2 u k ϕ dx, (4.7)
for every ϕ ∈ D s,p 0 (Ω). We now distinguish two cases. Case S p,s (a) = 0. If we consider the functional K introduced in (2.5) with µ = S p,s (a), recalling that u k has unit norm in L p * s (Ω), we obtain

| K (u k ), ϕ | = ˆR2N |u k (x) -u k (y)| p-2 (u k (x) -u k (y)) (ϕ(x) -ϕ(y)) |x -y| N +s p dx dy + ˆRN a |u k | p-2 u k ϕ dx -S p,s (a) ˆRN |u k | p * s -2 u k ϕ dx ≤ ˆRN |a k -a| |u k | p-1 |ϕ| dx + |S p,s (a) -S p,s (a k )| ˆRN |u k | p * s -1 |ϕ| dx ≤ a k -a L N/sp (Ω) + |S p,s (a) -S p,s (a k )| ϕ L p * s (R N ) .
This shows that {u k } k∈N is a Palais-Smale sequence for K at the level s/N S p,s (a). By recalling that S p,s (a) < S p,s , we obtain strong convergence in D s,p 0 (Ω) by Lemma 2.6. Case S p,s (a) = 0. From (4.5), we get

[u k ] p D s,p (R N ) + ˆRN a k |u k | p dx = o k (1),
thus by using the weak convergence of {|u k | p } k∈N and the strong convergence of {a k } k∈N , we obtain

[u k ] p D s,p (R N ) = - ˆRN a |u| p dx + o k (1).
On the other hand, by testing (4.7) with u and then taking the limit as k goes to ∞, we obtain By uniform convexity of the space D s,p 0 (Ω), we obtain the strong convergence in this case as well.

Proof of Theorem 1.2

We consider the two cases separately.

Case µ ≥ 0. We proceed by contradiction. Let us assume that for a µ ≥ 0 there exists a positive solution u 0 ∈ D s,p 0 (Ω) \ {0}. Thus u 0 satisfies ˆR2N J p (u 0 (x) -u 0 (y)) ϕ(x) -ϕ(y) |x -y| N +s p dx dy + We fix µ < 0, it is now sufficient to notice that v = t u, with t = µ S p,s (a)

1 p-p * s > 0,
is the seeked solution. Finally, we use Proposition 3.3 to infer its uniqueness.

Remark 5.1 (Negative potentials). For 1 ≤ q < p, let us define the sharp Poincaré constant for the embedding D s,p 0 (Ω) → L q (Ω), i.e.

λ p,q (Ω) = inf u∈D s,p 0 (Ω)

[u] p D s,p (R N ) :

ˆRN |u| q dx = 1 .
When the potential a is negative, i.e. when a + ≡ 0, then the condition S p,s (a) < 0 (and thus λ(Ω; a) < 1 by (3.6)) is verified if (5.1) ˆΩ a -q p-q dx p-q q < 1 λ p,q (Ω) .

Indeed, observe that by Hölder's inequality ˆRN |u| q dx ≤ ˆRN a -|u| p dx q p ˆΩ a -q p-q dx p-q p , 3 This test function is not admissible in principle, but it is sufficient to proceed as in the proof of Proposition 3.3.

We prefer to avoid these technicalities here.

where we used that p/q > 1. Thus by using this we get λ(Ω; a) = inf u∈D s,p 0 (Ω)\{0}

[u] p D s,p (R N ) ˆRN a -|u| p dx ≤ ˆΩ a -q p-q dx p-q q inf u∈D s,p 0 (Ω)\{0}

[u] p D s,p (R N ) ˆRN |u| q dx p q = ˆΩ a -q p-q dx p-q q λ p,q (Ω) < 1,

where the last estimate follows from (5.1). In the limit case q = p, we define the first eigenvalue of the fractional p-Laplacian λ 1 (Ω) = inf u∈D s,p 0 (Ω)

[u] p D s,p (R N ) :

ˆRN |u| p dx = 1 .
Then a sufficient condition for λ(Ω; a) < 1 to hold is inf Ω a -> λ 1 (Ω).

The proof is as above. Finally, if the potential a is a negative constant, i.e. a ≡ -λ with λ > 0, we observe that λ(Ω, a) := inf u∈D s,p 0 (Ω)

[u] p D s,p (R N ) :

ˆRN |u| p dx = 1 λ = inf u∈D s,p 0 (Ω) 1 λ [u] p D s,p (R N ) : ˆRN |u| p dx = 1 = λ 1 (Ω) λ ,
and condition (1.7) is equivalent to λ > λ 1 (Ω).

Proposition 2 . 2 (

 22 Discrete Picone inequality). Let 1 < p < ∞ and let a, b, c, d ∈ [0, +∞), with a, b > 0. Then J p (a -b) c p a p-1 -d p b p-1 ≤ |c -d| p . (2.3)Moreover, if equality holds in (2.3),

Lemma 3 . 1 .

 31 For every open set Ω ⊂ R N we have S p,s (0) = S p,s , and S p,s (0) is not attained in D s,p 0

  1), which yields λ n = S p,s (a) + o n (1). Therefore, a direct computation shows that { u n } n∈N ⊂ D s,p 0 (Ω) is a Palais-Smale sequence for the functional (2.5) with µ = S p,s (a) = 0, at the energy level c := s N S p,s (a).

Lemma 4 . 2 .

 42 The map T : L N/sp (Ω) → R defined byT (a) = S p,s (a), for a ∈ L N/sp (Ω), is 1-Lipschitz.In other words, for every a, a ∈ L N/sp (Ω) we have (4.4) S p,s (a) -S p,s (a ) ≤ a -a L N/sp (Ω) .

  [u] p D s,p (R N ) = -ˆRN a |u| p dx.The last two displays imply that lim k→∞[u k ] p D s,p (R N ) = [u] p D s,p (R N ) .

1

 1 We use the test function 3 ϕ = φ p 1 /u p-1 0 , where φ 1 is the positive solution of(3.4). This givesµ dx ≤ [φ 1 ] p D s,p (R N ) + ˆRN a + φ p 1 dx -ˆRN a -φ p 1 dx = λ(Ω; a) -1.The right-hand side is strictly negative by (1.7) and (3.6), thus we get a contradiction.Case µ < 0. Since we are assuming S p,s (a) < 0, by Proposition 3.4, we can infer that the variational problem defining S p,s (a) has a positive solution u ∈ D s,p 0 (Ω) \ {0}. As already observed, we have that u solves p ) s u + a u p-1 = S p,s (a) u p * s -1 , in Ω, u > 0,in Ω, u = 0, in R N \ Ω.

  .

	By the constrained version of Ekeland's variational principle (see [9, Theorem 3.1]) applied to the
	following functionals on D s,p 0 (Ω) J(u) := 1 p [u] p D s,p (R N ) +	1 p ˆRN	a |u| p dx	and	G(u) :=	1 p * s ˆRN	|u| p * s dx,
	there exist {λ

n } n∈N ⊂ R and { u n } n∈N ⊂ D s,p 0 (Ω) such that

The family {|u ε,δ | p }ε>0 is bounded in L p * s /p (Ω) and converges to 0 almost everywhere, by construction. Then we can apply [14, Lemme 4.8] again and infer vanishing of the term ´RN a |u ε,δ | p dx.


Pierre Bousquet

Lorenzo Brasco
email: lorenzo.brasco@unife.it

Chiara Leone
email: chiara.leone@unina.it

Anna Verde
email: anna.verde@unina.it

On the Lipschitz character of orthotropic p-harmonic functions

Keywords: 2010 Mathematics Subject Classification. 35J70, 35B65, 49K20 Degenerate elliptic equations, Anisotropic problems, Orthotropic problems, Lipschitz regularity

We prove that local weak solutions of the orthotropic p-harmonic equation are locally Lipschitz, for every p ≥ 2 and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces.

1. Introduction 1.1. The problem. In this paper, we pursue the study of the regularity of local minimizers of degenerate functionals with orthotropic structure, that we already considered in [START_REF] Bousquet | C 1 regularity of orthotropic p-harmonic functions in the plane[END_REF][START_REF] Bousquet | Lipschitz regularity for local minimizers of some widely degenerate problems[END_REF]3] and [START_REF] Brasco | Sobolev and Lipschitz regularity for local minimizers of widely degenerate anisotropic functionals[END_REF]. More precisely, for p ≥ 2, we consider local minimizers of the functional

(1.1) F 0 (u, Ω ) = N i=1 1 p ˆΩ |u x i | p dx, Ω Ω, u ∈ W 1,p loc (Ω ),
and more generally of the functional

F δ (u, Ω ) = N i=1 1 p ˆΩ (|u x i | -δ i ) p + dx + ˆΩ f u dx, Ω Ω, u ∈ W 1,p loc (Ω ).
Here, Ω ⊂ R N is an open set, N ≥ 2, and δ 1 , . . . , δ N are nonnegative numbers.

A local minimizer u of the functional F 0 defined in (1.1) is a local weak solution of the orthotropic p-Laplace equation (1.2)

N i=1 |u x i | p-2 u x i x i = 0.
For p = 2 this is just the Laplace equation, which is uniformly elliptic. For p > 2 this looks quite similar to the usual p-Laplace equation However, as explained in [START_REF] Bousquet | C 1 regularity of orthotropic p-harmonic functions in the plane[END_REF] and [START_REF] Bousquet | Lipschitz regularity for local minimizers of some widely degenerate problems[END_REF], equation (1.2) is much more degenerate. Consequently, as for the regularity of ∇u (i.e. boundedness and continuity), the two equations are dramatically different.

In order to understand this discrepancy between the p-Laplacian and its orthotropic version, let us observe that the map ξ → |ξ| p occuring in the definition (1.3) of I degenerates only at the origin, in the sense that its Hessian is positive definite on R N \ {0}. On the contrary, the definition of the orthotropic functional F 0 in (1.1) is related to the map ξ → N i=1 |ξ i | p , which degenerates on an unbounded set, namely the N hyperplanes orthogonal to the coordinate axes of R N .

The situation is even worse when (1.4) max{δ i : i = 1, . . . , N } > 0, for the lack of ellipticity of the degenerate p-orthotropic functional arises on the larger set

N i=1 {ξ ∈ R N : |ξ i | ≤ δ i }.
As a matter of fact, the regularity theory for these very degenerate functionals is far less understood than the corresponding theory for the standard case (1.3) and its variants. Under suitable integrability conditions on the function f , we can use the classical theory for functionals with p-growth and ensure that the local minimizers of F δ are locally bounded and Hölder continuous, see for example [8, Theorems 7. 5 & 7.6]. This theory also assures that the gradients of local minimizers lie in L r loc (Ω) for some r > p, see [8, Theorem 6.7]. We also point out that for f ∈ L ∞ loc (Ω), local minimizers of F δ are contained in W 1,q loc (Ω), for every q < +∞ (see [3,Main Theorem]).

1.2. Main result. In this paper, we establish the optimal regularity expected for the minimizers of F δ , namely the Lipschitz regularity 1 . More precisely, we establish the following result. 1 Observe that when f ≡ 0, any Lipschitz function u with |∇u| ≤ min{δi : i = 1, . . . , N } is a local minimizer of F δ . Thus in general Lipschitz continuity is the best regularity one can hope for.

Theorem 1.1. Let p ≥ 2, f ∈ W 1,h loc (Ω) for some h > N/2 and let U ∈ W 1,p loc (Ω) be a local minimizer of the functional F δ . Then U is locally Lipschitz in Ω.

Moreover, in the case δ 1 = • • • = δ N = 0, we have the following scaling invariant estimate: for every ball B 2R 0 Ω, it holds

(1.5) ∇U L ∞ (B R 0 /2 ) ≤ C B R 0 |∇U | p dx 1 p + C   R 2 0 B R 0 |∇f | h dx 1 h   1 p-1
, for some C = C(N, p, h) > 1.

Remark 1.2 (Comparison with previous results

). This result unifies and substantially extends the results on the orthotropic functional F δ contained in [START_REF] Bousquet | Lipschitz regularity for local minimizers of some widely degenerate problems[END_REF], where it has been established that the local minimizers of F δ are locally Lipschitz, provided that:

• p ≥ 2, N = 2 and f ∈ W 1,p loc (Ω), see [2, Theorem A]; • p ≥ 4, N ≥ 2 and f ∈ W 1,∞ loc (Ω), see [2, Theorem B].
The second result was based on the so-called Bernstein's technique, see for example [START_REF] Han | Elliptic partial differential equations[END_REF]Proposition 2.19]. This technique had already been exploited in the pioneering paper [11] by Uralt'seva and Urdaletova, for a class of functionals which contains the orthotropic functional F 0 defined in (1.1), but not its more degenerate version F δ . Namely, the result of [11] does not cover the case when condition (1.4) is in force.

Still for the case δ 1 = • • • = δ N = 0, an entirely different approach relying on viscosity methods has been developped in [START_REF] Demengel | Lipschitz interior regularity for the viscosity and weak solutions of the pseudo p-Laplacian equation[END_REF]. To our knowledge, both methods are limited to (at least) bounded lower order terms f . On the contrary, [2, Theorem A] can be considered as the true ancestor to Theorem 1.1 above. Indeed, they both follow the Moser's iteration technique, originally introduced in [10] to establish regularity for uniformly elliptic problems. However, going beyond the two-dimensional setting requires new ideas, that we will explain in Subsection 1.3 below.

In contrast to the partial results of [2, Theorems A & B], the proof of Theorem 1.1 does not depend on the dimension and does not need any additional restriction on p, apart from p ≥ 2. It allows unbounded lower order terms, even if the condition f ∈ W 1,h loc (Ω) for some h > N/2 is certainly not sharp. On this point, it is useful to observe that by Sobolev's embedding we have

W 1,h → L h * ,
with h * larger than N and as close to N as desired, provided h is close to N/2. This means that, in terms of summability, our assumption on f amounts to f ∈ L q loc (Ω) for some q > N . This is exactly the sharp expected condition on f for the local minimizers to be locally Lipschitz, at least if one nurtures the (optimistic) hope that the regularity for the orthotropic p-Laplacian agrees with that for the standard p-Laplacian.

Our strategy to prove Theorem 1.1 relies on energy methods and integral estimates, and more precisely on ad hoc Caccioppoli-type inequalities. This only requires growth assumptions on the Lagrangian and its derivatives and can be adapted to a large class of functionals. For instance, we briefly explain in Appendix A how to adapt our poof to the case of nonlinear lower order terms, i.e. when f u is replaced by a term of the form G(x, u).

Remark 1.3. We collect in this remark some interesting open issues:

(1) one word about the assumption p ≥ 2: as explained in [START_REF] Bousquet | C 1 regularity of orthotropic p-harmonic functions in the plane[END_REF] and [START_REF] Bousquet | Lipschitz regularity for local minimizers of some widely degenerate problems[END_REF], when

δ 1 = • • • = δ N = 0
the subquadratic case 1 < p < 2 is simpler in a sense. In this case, the desired Lipschitz regularity can be inferred from [6, Theorem 2.2] (see also [7, Theorem 2.7]). However, the more degenerate case (1.4) is open;

(2) in [1, Main Theorem], local minimizers were proven to be C 1 , in the two-dimensional case, for 1 < p < ∞ and when δ 1 = • • • = δ N = 0. We do not know whether such a result still holds in higher dimensions;

(3) in [START_REF] Brasco | Sobolev and Lipschitz regularity for local minimizers of widely degenerate anisotropic functionals[END_REF]Theorem 1.4], local Lipschitz regularity is established in the two-dimensional case for an orthotropic functional, with anisotropic growth conditions. Namely, for the functional

2 i=1 1 p i ˆ(|u x i | -δ i ) p i + dx + ˆf u dx, with 2 ≤ p 1 ≤ p 2 .
For such a functional, Lipschitz regularity is open in higher dimensions, even for the case However, steps b) and c) are quite involved, due to the degeneracy of our equation. This makes their concrete realization fairly intricate. Thus in order to smoothly introduce the reader to the proof, we prefer to spend some words. We point out that our proof is not just a mere adaption of techniques used for the p-Laplace equation. Moreover, it does not even rely on the ideas developed in [START_REF] Bousquet | Lipschitz regularity for local minimizers of some widely degenerate problems[END_REF] for the two-dimensional case. In a nutshell, we need new ideas to deal with our functional in full generality.

δ 1 = • • • = δ N = 0. 1.3.
In order to obtain "good" Caccioppoli-type inequalities for the gradient, we exploit an idea introduced in nuce in [START_REF] Bousquet | C 1 regularity of orthotropic p-harmonic functions in the plane[END_REF]. This consists in differentiating (1.2) in the direction x j and then testing the resulting equation with a test function of the form2

u x j |u x j | 2s-2 |u x k | 2m , with 1 ≤ s ≤ m.
This leads to an estimate of the type (see Proposition 4.1)

N i=1 ˆΩ |u x i | p-2 u 2 x i x j |u x j | 2 s-2 |u x k | 2 m dx ≤ C N i=1 ˆ|u x i | p-2 |u x j | 2 s+2 m + |u x k | 2 s+2 m dx + N i=1 ˆ|u x i | p-2 u 2 x i x j |u x j | 4 s-2 |u x k | 2 m-2 s dx. (1.6)
Then the idea is the following: let us suppose that we are interested in improving the summability of the component u x k . Ideally, we would like to take s = 1 in (1.6), since in this case the left-hand side boils down to

N i=1 ˆ|u x i | p-2 u 2 x i x j u 2 m x k dx ≥ ˆ|u x k | p-2 u 2 x k x j u 2 m x k dx ˆ |u x k | p 2 +m x j 2
dx.

If we now sum over j = 1, . . . , N , this would give a control on the W 1,2 norms of convex powers of u x k . But there is a drawback here: indeed, this W 1,2 norm is estimated still in terms of the Hessian of u, which is contained in the right-hand side of (1.6). Observe that (1.6) has the following form

(1.7) I(s -1, m) ≤ C N i=1 ˆ|u x i | p-2 |u x j | 2 s+2 m + |u x k | 2 s+2 m dx + I(2 s -1, m -s), where
I(s, m) = N i=1 ˆΩ |u x i | p-2 u 2 x i x j |u x j | 2 s |u x k | 2 m dx.
This suggests to perform a finite iteration of (1.7) for s = s i and m = m i such that

2 s i -1 = s i+1 -1 s 0 = 1 and m i -s i = m i+1 , for i = 0, . . . , .
The number is chosen so that we stop the iteration when we reach m = 0. The above conditions imply that for every i = 0, . . . , , we have

m i + s i = m 0 + s 0 = 2 .
In this way, after a finite number of steps (comparable to 2 ), the coupling between u x k and the Hessian of u contained in the term I will disappear from the right-hand side. In other words, we will end up with an estimate of the type

ˆ ∇|u x k | 2 + p-2 2 2 dx ≤ C N i,j=1 ˆ|u x i | p-2 |u x j | 2 +1 + |u x k | 2 +1 dx + N i=1 ˆ|u x i | p-2 u 2 x i x j u 2 (2 -1)
x j dx.

(1.8)

Observe that we still have the Hessian of u in the right-hand side (this is the second term), but this time it is harmless. It is sufficient to use the standard Caccioppoli inequality (3.3) for the gradient, which reads

N i=1 ˆ|u x i | p-2 u 2 x i x j u 2 (2 -1) x j dx N i=1 ˆ|u x i | p-2 u 2 +1
x j dx, and the last term is already contained in the right-hand side of (1.8). All in all, by applying Sobolev inequality in the left-hand side of (1.8), we get the following type of self-improving information

∇u ∈ L 2 γ (B R ) =⇒ ∇u ∈ L 2 * γ (B r ), where we set γ = p -2 2 + 2 .
In this way, we obtain an iterative scheme of reverse Holder's inequalities. This is Step 1 in the proof of Proposition 5.1 below. Thus, apparently, we safely landed in step c) of the strategy described above.

We now want to pass to step d) and iterate infinitely many times the previous information. The goal would be to define the diverging sequence of exponents γ by

γ = p -2 2 + 2 , ≥ 1,
and conclude by iterating

(1.9) ∇u ∈ L 2 γ (B R ) =⇒ ∇u ∈ L 2 * γ (B r ).
Once again, there is a drawback. Indeed, observe that by definition 2 * 2 γ = γ +1 .

One may think that this is not a big issue: indeed, it would be sufficient to have

(1.10) γ +1 ≤ 2 * 2 γ ,
then an application of Hölder's inequality in (1.9) would lead us to

∇u ∈ L 2 γ (B R ) =⇒ ∇u ∈ L 2 γ +1 (B r ),
and we could enchain all the estimates. However, since the ratio 2 * /2 tends to 1 as the dimension N goes to ∞, it is easy to see that (1.10) cannot be true in general. More precisely, such a condition holds only up to dimension N = 4. The idea is then to go back to (1.9) and use interpolation in Lebesgue spaces in order to construct a Moser's scheme "without holes". In a nutshell, we interpolate the term

ˆBR |∇u| 2 γ dx, between ˆBR |∇u| 2 γ -1 dx and ˆBR |∇u| 2 * γ dx,
and use an iteration over shrinking radii in order to absorb the last term, see Step 2 of the proof of Proposition 5.1. Once this is done, we end up with the updated self-improving information

∇u ∈ L 2 γ -1 (B R ) =⇒ ∇u ∈ L 2 * γ (B r ).
What we gain is that now 2 * γ > 2 γ > 2 γ -1 , thus by using Hölder's inequality we obtain

∇u ∈ L 2 γ -1 (B R ) =⇒ ∇u ∈ L 2 γ (B r ).
The information comes with a precise iterative estimate and a good control on the relevant constants. We can thus launch the Moser's iteration procedure and obtain the desired L ∞ estimate, see Step 3 of the proof of Proposition 5.1.

There is still a small detail that needs some care: the first exponent of the iteration is

2 γ 0 = p + 2,
which means that on ∇u we obtain a L ∞ -L p+2 local estimate. Finally, in order to obtain the desired L ∞ -L p estimate, one can simply use an interpolation argument (this is Step 4 of the proof of Proposition 5.1).

1.4. Plan of the paper. In Section 2, we define the approximation scheme and settle all the needed machinery. We have dedicated Section 3 to the new Caccioppoli inequalities which mix together the derivatives of the gradient with respect to 2 orthogonal directions. In Section 4, we exploit these Caccioppoli inequalities to establish integrability estimates on power functions of the gradient. In the subsequent section, we rely on these estimates to construct a Moser's iteration scheme which finally leads to the uniform a priori estimate of Proposition 5.1.

For ease of readability, both in Sections 4 and 5, we first consider the case f = 0 and δ = 0, in order to emphasize the main ideas and novelties of our approach. We explain subsequently in Subsections 4.2 and 5.2 respectively the technicalities to cover the general case f ∈ W 1,h loc (Ω) and max{δ

i : i = 1, . . . , N } > 0.
Finally, in Appendix A, we generalize Theorem 1.1 to nonlinear lower order terms.

Acknowledgments. We thank Nicola Fusco for his encouragements and some useful discussions.

Preliminaries

We will use the same approximation scheme as in [2, Section 2]. We introduce the notation

g i (t) = 1 p (|t| -δ i ) p + , t ∈ R, i = 1, . . . , N,
where 0 ≤ δ 1 , . . . , δ N are given real numbers and we also set

(2.1) δ = 1 + max{δ i : i = 1, . . . , N }.
We are interested in local minimizers of the following variational integral

F δ (u; Ω ) = N i=1 ˆΩ g i (u x i ) dx + ˆΩ f u dx, u ∈ W 1,p loc (Ω),
where Ω Ω and f ∈ W 1,h loc (Ω) for some h > N/2. The latter implies that

f ∈ L h * loc (Ω) ⊂ L N loc (Ω) ⊂ L p loc (Ω)
. The last inclusion is a consequence of the fact that p ≥ 2 and N ≥ 2. The condition f ∈ L p loc is exactly the one required in [2, Section 2] to justify the approximation scheme that we now describe.

We set

(2.2) g i,ε (t) = g i (t) + ε 2 t 2 = 1 p (|t| -δ i ) p + + ε 2 t 2 , t ∈ R.
Remark 2.1. For p = 2 and δ i > 0, we have

g i ∈ C 1,1 (R) ∩ C ∞ (R \ {δ i , -δ i }), but g i is not C 2 .
In this case, like in [3, Section 2] one would need to replace g i by a regularized version, in particular for the C 2 regularity result of Lemma 2.2 below. In order not to overburden the presentation, we prefer to avoid to explicitely write down this regularization and keep on using the same symbol g i .

From now on, we fix U a local minimizer of F δ . We also fix a ball

B Ω such that 2 B Ω as well.

Here λ B denotes the ball having the same center as B, scaled by a factor λ > 0. For every 0 < ε 1 and every x ∈ B, we set

U ε (x) = U * ε (x)
, where ε is a smooth convolution kernel, supported in a ball of radius ε centered at the origin.

Finally, we define

F δ,ε (v; B) = N i=1 ˆB g i,ε (v x i ) dx + ˆB f ε v dx,
where f ε = f * ε . The following preliminary result is standard, see [2, Lemma 2.5 and Lemma 2.8].

Lemma 2.2 (Basic energy estimate).

There exists ε 0 > 0 such that for every 0 < ε ≤ ε 0 < 1, the problem

(2.3) min F ε (v; B) : v -U ε ∈ W 1,p 0 (B)
, admits a unique solution u ε . Moreover, there exists a constant C = C(N, p) > 0 such that the following uniform estimate holds

ˆB |∇u ε | p dx ≤ C ˆ2 B |∇U | p dx + |B| p N ˆ2 B |f | p dx + (ε 0 + (δ -1) p )|B| . Finally, u ε ∈ C 2 (B).
We also rely on the following stability result, which is slightly more precise than [2, Lemma 2.9].

Lemma 2.3 (Convergence to a minimizer).

With the same notation as before, there exists a sequence {ε k } k∈N ⊂ (0, ε 0 ) converging to 0, such that

lim k→∞ u ε k -u L p (B) = 0, where u is a solution of min F δ (v; B) : v -U ∈ W 1,p 0 (B) .
We also have

(2.4) u x i -U x i ≤ 2 δ i , for a. e. x ∈ B, i = 1, . . . , N.
In the case δ = 1, i.e. when δ 1 = • • • = δ N = 0, then u = U and we have the stronger convergence

(2.5) lim k→∞ u ε k -U W 1,p (B) = 0.
Proof. The first part is proven in [2, Lemma 2.9], while (2.4) is proven in [2, Lemma 2.3]. For the case δ = 1, we observe that u = U follows from the strict convexity of the functional, together with the local minimality of U . In order to prove (2.5), we observe that

N i=1 1 p ˆB |(u ε k ) x i | p dx - N i=1 ˆB |U x i | p dx ≤ |F δ,ε k (u ε k ) -F δ (U )| + ε k 2 ˆB |∇u ε k | 2 dx + ˆB f ε k u ε k dx -ˆB f U dx .
We now use that {u ε k } k∈N strongly converges in L p (B), is bounded in W 1,p (B) and that {f ε k } k∈N strongly converges in L p (B) to f . By further using that

lim k→∞ |F δ,ε k (u ε k ) -F δ (U )| = 0, we finally get lim k→∞ N i=1 ˆB |(u ε k ) x i | p dx = N i=1 ˆB |U x i | p dx, i = 1, . . . , N.
This and the uniform convexity of L p (B) gives (2.5).

Remark 2.4. Observe that the functional F δ is not strictly convex when δ > 1. Thus property (2.4) is useful in order to transfer a Lipschitz estimate for the minimizer u selected in the limit, to the chosen one U .

Finally, we will repeatedly use the following classical result, see [8, Lemma 6.1] for a proof.

Lemma 2.5. Let 0 < r < R and let Z(t) : [r, R] → [0, ∞) be a bounded function. Assume that for r ≤ t < s ≤ R we have Z(t) ≤ A (t -s) α 0 + B (t -s) β 0 + C + ϑ Z(s), with A, B, C ≥ 0, α 0 ≥ β 0 > 0 and 0 ≤ ϑ < 1. Then we have Z(r) ≤ 1 (1 -λ) α 0 λ α 0 λ α 0 -ϑ A (R -r) α 0 + B (R -r) β 0 + C ,
where λ is any number such that

ϑ 1 α 0 < λ < 1.

Caccioppoli-type inequalities

The solution u ε of the regularized problem (2.3) satisfies the Euler-Lagrange equation

(3.1) N i=1 ˆg i,ε ((u ε ) x i ) ϕ x i dx + ˆfε ϕ dx = 0, ϕ ∈ W 1,p 0 (B).
From now on, in order to simplify the notation, we will systematically forget the subscript ε on u ε and f ε and simply write u and f respectively. We now insert a test function of the form ϕ = ψ x j ∈ W 1,p 0 (B) in (3.1), compactly supported in B. Then an integration by parts yields

(3.2) N i=1 ˆg i,ε (u x i ) u x i x j ψ x i dx + ˆfx j ψ dx = 0,
for j = 1, . . . , N . This is the equation solved by u x j .

We refer to [2, Lemma 3.2] for a proof of the following Caccioppoli inequality: Lemma 3.1. Let Φ : R → R + be a C 1 convex function. Then there exists a constant C = C(p) > 0 such that for every function η ∈ C ∞ 0 (B) and every j = 1, . . . , N , we have

N i=1 ˆg i,ε (u x i ) Φ(u x j ) x i 2 η 2 dx ≤ C N i=1 ˆg i,ε (u x i ) |Φ(u x j )| 2 η 2 x i dx + C ˆ|f x j | |Φ (u x j )| |Φ(u x j )| η 2 dx. (3.3)
We need a more elaborate Caccioppoli-type inequality, which is reminiscent of [1, Proposition 3.1]. Proposition 3.2 (Weird Caccioppoli inequality). Let Φ, Ψ : [0, +∞) → [0, +∞) be two nondecreasing continuous functions. We further assume that Ψ is convex and

C 1 . Let η ∈ C ∞ 0 (B) and 0 ≤ θ ≤ 2, then for every k, j = 1, . . . , N , N i=1 ˆg i,ε (u x i ) u 2 x i x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 dx ≤ C N i=1 ˆg i,ε (u x i ) u 2 x j Φ(u 2 x j ) Ψ(u 2 x k ) |∇η| 2 dx + C N i=1 ˆg i,ε (u x i ) u 2 x i x j u 2 x j Φ(u 2 x j ) 2 Ψ (u 2 x k ) θ η 2 dx 1 2 ×   N i=1 ˆg i,ε (u x i ) |u x k | 2θ Ψ(u 2 x k ) 2-θ |∇η| 2 dx 1 2 + E 1 (f ) 1 2   + C E 2 (f ) (3.4)
where

E 1 (f ) := ˆ|f x k | |u x k | θ+1 Ψ(u 2 x k ) Ψ (u 2 x k ) 1-θ 2 η 2 dx, E 2 (f ) := ˆ|f x j | |u x j | Φ(u 2 x j ) Ψ(u 2 x k ) η 2 dx.
Proof. By a standard approximation argument, one can assume that Φ and Ψ are C 1 . We take in (3.2)

ϕ = u x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 .
This gives

N i=1 ˆg i,ε (u x i ) u 2 x i x j Φ(u 2 x j ) + 2u 2 x j Φ (u 2 x j ) Ψ(u 2 x k ) η 2 dx = -2 N i=1 ˆg i,ε (u x i ) u x i x j u x j Φ(u 2 x j ) Ψ(u 2 x k ) η η x i dx -2 N i=1 ˆg i,ε (u x i ) u x i x j u x j u x i x k u x k Ψ (u 2 x k ) Φ(u 2 x j ) η 2 dx -ˆfx j u x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 dx =: A 1 + A 2 + A 3 . (3.5)
We now proceed to estimating the three terms A . We have

A 1 ≤ 1 2 N i=1 ˆg i,ε (u x i ) u 2 x i x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 dx + 2 N i=1 ˆg i,ε (u x i ) u 2 x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 x i dx
and the Hessian of u can be absorbed in the left-hand side of (3.5). Using also that 2 u 2

x j Φ (u 2 x j ) ≥ 0, this yields 1 2 N i=1 ˆΩg i,ε (u x i ) u 2 x i x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 dx ≤ 2 N i=1 ˆg i,ε (u x i ) u 2 x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 x i dx + A 2 + A 3 .
(3.6)

We now estimate A 2 , which is the most delicate term: writing Ψ (u

2 x k ) = Ψ (u 2 x k ) θ 2 Ψ (u 2
x k ) 1-θ 2 and using Cauchy-Schwarz inequality, we get

A 2 ≤ 2 N i=1 ˆg i,ε (u x i ) u 2 x i x j u 2 x j Φ(u 2 x j ) 2 Ψ (u 2 x k ) θ η 2 dx 1 2 × N i=1 ˆg i,ε (u x i ) u 2 x i x k u 2 x k Ψ (u 2 x k ) 2-θ η 2 dx 1 2
.

(3.7)

We observe that

N i=1 ˆg i,ε (u x i ) u 2 x i x k u 2 x k Ψ (u 2 x k ) 2-θ η 2 = 1 4 N i=1 ˆg i,ε (u x i ) (G(u x k )) x i 2 η 2 ,
where G is the convex nonnegative C 1 function defined by

G(t) = ˆt2 0 Ψ (τ ) 1-θ 2 dτ.
Thus by Caccioppoli inequality (3.3), we get

N i=1 ˆg i,ε (u x i ) u 2 x i x k u 2 x k Ψ (u 2 x k ) 2-θ η 2 ≤ C N i=1 ˆg i,ε (u x i ) G(u x k ) 2 η 2 x i dx + C ˆ|f x k | G(u x k ) G (u x k ) η 2 dx. By Jensen's inequality 0 ≤ G(u x k ) ≤ |u x k | θ ˆu2 x k 0 Ψ (τ ) dτ 1-θ 2 ≤ |u x k | θ Ψ(u 2 x k ) 1-θ 2 .
Together with the fact that G (u

x k ) = 2 u x k Ψ (u 2 x k ) 1-θ 2 , this implies N i=1 ˆg i,ε (u x i ) u 2 x i x k u 2 x k Ψ (u 2 x k ) 2-θ η 2 ≤ C N i=1 ˆg i,ε (u x i ) |u x k | 2θ Ψ(u 2 x k ) 2-θ η 2 x i dx + C ˆ|f x k | |u x k | θ+1 Ψ(u 2 x k ) Ψ (u 2 x k ) 1-θ 2 η 2 dx,
which in turn yields by (3.6) and (3.7),

1 2 N i=1 ˆΩg i,ε (u x i ) u 2 x i x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 dx ≤ 2 N i=1 ˆg i,ε (u x i ) u 2 x j Φ(u 2 x j ) Ψ(u 2 x k ) η 2 x i dx + C N i=1 ˆg i,ε (u x i ) u 2 x i x j u 2 x j Φ(u 2 x j ) 2 Ψ (u 2 x k ) θ η 2 dx 1 2 ×   N i=1 ˆg i,ε (u x i ) |u x k | 2θ Ψ(u 2 x k ) 2-θ η 2 x i dx 1 2 + ˆ|f x k | |u x k | θ+1 Ψ(u 2 x k ) Ψ (u 2 x k ) 1-θ 2 η 2 dx 1 2 + A 3 .
Here, we have also used the inequality

(A + B) 1/2 ≤ A 1/2 + B 1/2 .
Finally,

A 3 ≤ C ˆΩ |f x j | |u x j | Φ(u 2 x j ) Ψ(u 2 x k ) η 2 dx.
This completes the proof.

Local energy estimates for the regularized problem

In order to emphasize the main ideas of the proof, we have divided this section in two parts. In the first one, we explain how (3.4) leads to higher integrability estimates for the gradient when f = 0 and δ = 1. This allows to ignore a certain amount of technicalities. In the second part, we then detail the modifications of the proof to obtain the corresponding estimates in the general case.

4.1. The homogeneous case. In this subsection, we assume that f = 0 and δ = 1. Then the two terms E 1 (f ) and E 2 (f ) in (3.4) vanish. Also observe that in this case from (2.2) we have

g i,ε (t) = (p -1) |t| p-2 + ε.
Let us single out a particular case by taking

≤ s ≤ m N i=1 ˆΩ g i,ε (u x i ) u 2 x i x j |u x j | 2 s-2 |u x k | 2 m η 2 dx ≤ C N i=1 ˆg i,ε (u x i ) |u x j | 2 s+2 m |∇η| 2 dx + C (m + 1) N i=1 ˆg i,ε (u x i ) |u x k | 2 s+2 m |∇η| 2 dx + N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x j | 4 s-2 |u x k | 2 m-2 s η 2 dx. (4.2)
Proof. We use (3.4) with the choices (4.1) above and

θ =        m -s m -1 ∈ [0, 1] if m > 1, 1 if m = 1.

This gives

N i=1 ˆΩg i,ε (u x i ) u 2 x i x j |u x j | 2 s-2 |u x k | 2 m η 2 dx ≤ C N i=1 ˆg i,ε (u x i ) |u x j | 2 s |u x k | 2 m |∇η| 2 dx + C m θ N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x j | 4 s-2 |u x k | 2 m-2 s η 2 dx 1 2 × N i=1 ˆΩ g i,ε (u x i ) |u x k | 2 m+2 s |∇η| 2 dx 1 2
.

We use Young's inequality in the form C √ a b ≤ C 2 b/4 + a for the product in the right-hand side to get

N i=1 ˆΩg i,ε (u x i ) u 2 x i x j |u x j | 2 s-2 |u x k | 2 m η 2 dx ≤ C N i=1 ˆg i,ε (u x i ) |u x j | 2 s |u x k | 2 m |∇η| 2 dx + C m θ N i=1 ˆΩ g i,ε (u x i ) |u x k | 2 m+2 s |∇η| 2 dx + N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x j | 4 s-2 |u x k | 2 m-2 s η 2 dx.
In the first term of the right-hand side, we use Young's inequality with the exponents

2 m + 2 s 2 s , 2 m + 2 s 2 m .
We also observe for the second term that m θ ≤ m. This gives the desired estimate.

Proposition 4.2 (Caccioppoli for power functions of the gradient). We fix an exponent q = 2 0 -1, for a given 0 ∈ N \ {0}.

Let η ∈ C ∞ 0 (B), then for every k = 1, . . . , N we have

ˆ ∇ |u x k | q+ p-2 2 u x k 2 η 2 dx ≤ C q 5 N i,j=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx + C q 5 N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx, (4.3)
for some C = C(N, p) > 0.

Proof. We define the two finite families of indices {s } and {m } such that

s = 2 , m = q + 1 -2 , ∈ {0, . . . , 0 }.
Observe that s + m = q + 1, ∈ {0, . . . , 0 },

and 4 s -2 = 2 s +1 -2, 2 m -2 s = 2 m +1 , and
s 0 = 1, m 0 = q, s 0 = 2 0 , m 0 = 0.
In terms of these families, inequality (4.2) implies

N i=1 ˆΩg i,ε (u x i ) u 2 x i x j |u x j | 2 s -2 |u x k | 2 m η 2 dx ≤ C N i=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx + C (m + 1) N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx + N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x j | 4 s -2 |u x k | 2 m -2 s η 2 dx,
for some C > 0 universal. By starting from = 0 and iterating up to = 0 -1 so that 1 ≤ s ≤ m , we then get

N i=1 ˆΩg i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≤ C q 2 N i=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx + C q 2 N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx + N i=1 ˆg i,ε (u x i ) u 2 x i x j u 2 q x j η 2 dx,
for a universal constant C > 0. For the last term, we apply the Caccioppoli inequality (3.3) for the gradient, thus we get

N i=1 ˆΩ g i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≤ C q 2 N i=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx + C q 2 N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx + C (q + 1) 2 N i=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx; that is, N i=1 ˆΩ g i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≤ C q 2 N i=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx + C q 2 N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx. (4.4) We now observe that g i,ε (u x i ) = (p -1) |u x i | p-2 + ε and thus N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≥ ˆ|u x k | p-2 u 2 x k x j |u x k | 2 q η 2 dx = 2 2 q + p 2 ˆ |u x k | q+ p-2 2 u x k x j 2 η 2 dx,
thus when we sum over j = 1, . . . , N we get N i,j=1

ˆ|u x i | p-2 u 2 x i x j |u x k | 2 q η 2 dx ≥ 2 2 q + p 2 ˆ ∇ |u x k | q+ p-2 2 u x k 2 η 2 dx.
This proves the desired inequality.

4.2.

The non-homogeneous case. In the general case where f = 0 and/or δ > 1, we can prove the following analogue of (4.2), in a similar way:

N i=1 ˆΩ g i,ε (u x i ) u 2 x i x j |u x j | 2 s-2 |u x k | 2 m η 2 dx ≤ N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x j | 4 s-2 |u x k | 2 m-2 s η 2 dx + C m N i=1 ˆg i,ε (u x i ) (|u x j | 2 (s+m) + |u x k | 2 (s+m) ) |∇η| 2 dx + C m 2 ˆ|∇f | (|u x k | 2 (s+m)-1 + |u x j | 2 (s+m)-1 ) η 2 dx. (4.5)
We then deduce the following analogue of Proposition 4.2:

Proposition 4.3. We fix an exponent q = 2 0 -1, for a given 0 ∈ N \ {0}.

Let η ∈ C ∞ 0 (Ω), then for every k = 1, . . . , N we have

ˆ ∇ (|u x k | -δ k ) p 2 + |u x k | q 2 η 2 dx ≤ C q 5 N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 + N j=1 |u x j | 2 q+2 |∇η| 2 dx + C q 5 ˆ|∇f | |u x k | 2 q+1 + N j=1 |u x j | 2 q+1 η 2 dx, (4.6)
for some C = C(N, p) > 0.

Proof. Using the same notation and the same strategy as in the proof of (4.3), except that we start from (4.5) instead of (4.2), we get the following analogue of (4.4):

N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≤ C q 2 N i=1 ˆg i,ε (u x i ) (|u x j | 2 q+2 + |u x k | 2 q+2 ) |∇η| 2 dx + C q 3 ˆ|∇f | (|u x k | 2 q+1 + |u x j | 2 q+1 ) η 2 dx.
We now observe that

N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≥ (p -1) ˆ(|u x k | -δ k ) p-2 + u 2 x k x j |u x k | 2 q η 2 dx.

Noting that (|u

x k | -δ k ) p + ≤ (|u x k | -δ k ) p-2 + |u x k | 2 , we have (|u x k | -δ k ) p 2 + |u x k | q x j 2 ≤ 2 (|u x k | -δ k ) p 2 + x j 2 |u x k | 2 q + 2 (|u x k | -δ k ) p + (|u x k | q ) x j 2 ≤ C q 2 (|u x k | -δ k ) p-2 + |u x k | 2 q u 2 x k x j . We deduce therefrom N i=1 ˆg i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≥ C q 2 ˆ (|u x k | -δ k ) p 2 + |u x k | q x j 2 η 2 dx,
thus when we sum over j = 1, . . . , N, we get

N i,j=1 ˆg i,ε (u x i ) u 2 x i x j |u x k | 2 q η 2 dx ≥ C q 2 ˆ ∇ (|u x k | -δ k ) p 2 + |u x k | q 2 η 2 dx.
This proves the desired inequality (4.6).

Proof of Theorem 1.1

Proof. The core of the proof of Theorem 1.1 is the uniform Lipschitz estimate of Proposition 5.1 below. Its proof, which is postponed for ease of readability, uses the integrability estimates of Section 4. Once we have this uniform estimate, we can reproduce the proof of [2, Theorem A] and prove that ∇U ∈ L ∞ (Ω ), for every Ω Ω. We now detail how to obtain the scaling invariant local estimate (1.5) in the case δ 1 = • • • = δ N = 0. We take 0 < r 0 < R 0 ≤ 1 and a ball B 2R 0 Ω. We then consider the sequence of miminizers {u ε k } k∈N of (2.3) obtained in Lemma 2.3, with B a ball slightly larger than B R 0 so that 2 B Ω. By using the uniform Lipschitz estimate (5.3) below, taking the limit as k goes to ∞ and using the strong convergence of Lemma 2.3, we obtain

∇U L ∞ (Br 0 ) ≤ C (R 0 -r 0 ) σ 2 1 + ∇f σ 2 L h (B R 0 ) ∇U σ 1 L p (B R 0 ) + 1 .
Without loss of generality, we can assume that ∇U L p (B R 0 ) > 0. Hence, by Young's inequality, (5.1)

∇U L ∞ (Br 0 ) ≤ C (R 0 -r 0 ) σ 2 1 + ∇f 2 σ 2 L h (B R 0 ) + ∇U 2 σ 1 L p (B R 0 ) ,
possibly for a different C = C(N, p, h) > 0. We now observe that for every λ > 0, λ U is still a solution of the orthotropic p-Laplace equation, with the right hand side f replaced by λ p-1 f . We can use (5.1) for λ U and get

λ ∇U L ∞ (Br 0 ) ≤ C (R 0 -r 0 ) σ 2 1 + λ 2 σ 2 (p-1) ∇f 2 σ 2 L h (B R 0 ) + λ 2 σ 1 ∇U 2 σ 1 L p (B R 0 ) .
Dividing by λ, we obtain

∇U L ∞ (Br 0 ) ≤ C (R 0 -r 0 ) σ 2 1 λ + λ 2 σ 2 (p-1)-1 ∇f 2σ 2 L h (B R 0 ) + λ 2 σ 1 -1 ∇U 2 σ 1 L p (B R 0 ) .
We take

λ := 1 ∇U L p (B R 0 ) + ∇f 1 p-1 L h (B R 0 ) , and observe that if ∇f L h (B R 0 ) > 0, then λ 2 σ 2 (p-1)-1 ∇f 2σ 2 L h (B R 0 ) ≤ 1 ∇f 1 p-1 L h (B R 0 ) 2 σ 2 (p-1)-1 ∇f 2σ 2 L h (B R 0 ) = ∇f 1 p-1 L h (B R 0 )
while the inequality is obvious when ∇f L h (B R 0 ) = 0. Similarly,

λ 2 σ 1 -1 ∇U 2 σ 1 L p (B R 0 ) ≤ 1 ∇U 2 σ 1 -1 L p (B R 0 ) ∇U 2 σ 1 L p (B R 0 ) = ∇U L p (B R 0 ) .
It thus follows that

(5.2) ∇U L ∞ (Br 0 ) ≤ C (R 0 -r 0 ) σ 2 ∇f 1 p-1 L h (B R 0 ) + ∇U L p (B R 0 ) .
We now make this estimate dimensionally correct. Given R 0 > 0, we consider a ball B 2R 0 Ω. Then the rescaled function

U R 0 (x) = U (R 0 x), for x ∈ R -1 0 Ω,
is a solution of the orthotropic p-Laplace equation, with right-hand side f R 0 (x) := R p 0 f (R 0 x). We can use for it the estimate (5.2) with radii 1 and 1/2. By scaling back, we thus obtain

R 0 ∇U L ∞ (B R 0 /2 ) ≤ C R -N p +1 0 ∇U L p (B R 0 ) + R h (p+1)-N h (p-1) 0 ∇f 1 p-1 L h (B R 0 ) , for some constant C = C(N, p, h) > 1. Dividing by R 0 , we get ∇U L ∞ (B R 0 /2 ) ≤ C B R 0 |∇U | p dx 1 p + C R 2 p-1 -N h (p-1) 0 ˆBR 0 |∇f | h dx 1 h (p-1)
. This concludes the proof.

Proposition 5.1 (Uniform Lipschitz estimate). Let p ≥ 2, h > N/2 and 0 < ε ≤ ε 0 . For every B r 0 ⊂ B R 0 B with 0 < r 0 < R 0 ≤ 1, we have (5.3) ∇u ε L ∞ (Br 0 ) ≤ C 1 + ∇f ε σ 2 L h (B R 0 ) (R 0 -r 0 ) σ 2 ∇u ε σ 1 L p (B R 0 ) + 1 ,
where C = C(N, p, h, δ) > 1 and σ i = σ i (N, p, h) > 0, for i = 1, 2.

5.1. Proof of Proposition 5.1: the homogeneous case. In this subsection, we assume that f = 0 and δ = 1.

For simplicity, we assume throughout the proof that N ≥ 3, so in this case the Sobolev exponent 2 * is well-defined. The case N = 2 can be treated with minor modifications. For ease of readability, we divide the proof into four steps.

Step 1: a first iterative scheme. We add on both sides of inequality (4.3) the term

ˆ|∇η| 2 |u x k | 2 q+p dx.
We thus obtain

ˆ ∇ |u x k | q+ p-2 2 u x k η 2 dx ≤ C q 5 N i,j=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx + C q 5 N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx + C ˆ|∇η| 2 |u x k | 2 q+p dx.
An application of Sobolev inequality leads to

ˆ|u x k | 2 * 2 (2 q+p) η 2 * dx 2 2 * ≤ C q 5 N i,j=1 ˆg i,ε (u x i ) |u x j | 2 q+2 |∇η| 2 dx + C q 5 N i=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx + C ˆ|∇η| 2 |u x k | 2 q+p dx.
We now sum over k = 1, . . . , N and use that by Minkowski inequality,

N k=1 ˆ|u x k | 2 * 2 (2 q+p) η 2 * dx 2 2 * = N k=1 |u x k | 2 q+p η 2 L 2 * 2 ≥ N k=1 |u x k | 2 q+p η 2 L 2 * 2 .
This implies

   ˆ N k=1 |u x k | 2 q+p 2 * 2 η 2 * dx    2 2 * ≤ C q 5 N i,k=1 ˆg i,ε (u x i ) |u x k | 2 q+2 |∇η| 2 dx + C ˆ|∇η| 2 N k=1 |u x k | 2 q+p dx. (5.4)
We now introduce the function U(x) := max k=1,...,N

|u x k (x)|.
We use that

U 2 q+p ≤ N k=1 |u x k | 2 q+p ≤ N U 2 q+p ,
and also that g i,ε (u

x i ) |u x k | 2 q+2 ≤ C U 2 q+p + ε U 2 q+2 for every 1 ≤ i, k ≤ N . This yields ˆU 2 * 2 (2 q+p) η 2 * 2 2 * ≤ C q 5 ˆU2 q+p |∇η| 2 dx + Cq 5 ε ˆU2q+2 |∇η| 2 dx
for a possibly different C = C(N, p) > 1. By using that U 2 q+2 ≤ 1 + U 2 q+p , we obtain (for ε < 1)

(5.5) ˆU 2 * 2 (2 q+p) η 2 * dx 2 2 * ≤ C q 5 ˆ|∇η| 2 U 2q+p + 1 dx.
We fix two concentric balls B r ⊂ B R B and 0 < r < R ≤ 1. Let us assume for simplicity that all the balls are centered at the origin. Then for every pair of radius r ≤ t < s ≤ R we take in (5.5) the cut-off function

(5.6) η(x) = min 1, (s -|x|) + s -t .
This yields

(5.7) ˆBt U 2 * 2 (2 q+p) dx 2 2 * ≤ C q 5 (s -t) 2 ˆBs U 2 q+p + 1 dx.
We define the sequence of exponents

γ j = p + 2 j+2 -2, j ∈ N,
and take in (5.7) q = 2 j+1 -1. This gives ˆBt U

2 * 2 γ j dx 2 2 * ≤ C 2 5 j (s -t) 2 ˆBs U γ j + 1 dx, (5.8) for a possibly different constant C = C(N, p) > 1.
Step 2: filling the gaps. We now observe that

γ j-1 < γ j < 2 * 2 γ j , for every j ∈ N \ {0}.
By interpolation in Lebesgue spaces, we obtain ˆBt

U γ j dx ≤ ˆBt U γ j-1 dx τ j γ j γ j-1 ˆBt U 2 * 2 γ j dx (1-τ j ) 2 2 *
where 0 < τ j < 1 is given by

τ j = 2 * 2 -1 2 * 2 γ j γ j-1 - 1
.

We now rely on (5.8) to get ˆBt

U γ j dx ≤ ˆBt U γ j-1 dx τ j γ j γ j-1 C 2 5 j (s -t) 2 ˆBs U γ j + 1 dx 1-τ j =   C 2 5 j (s -t) 2 1-τ j τ j ˆBt U γ j-1 dx γ j γ j-1   τ j ˆBs U γ j + 1 dx 1-τ j .
The sequence (τ j ) j≥1 is decreasing, which implies

τ j > lim n→∞ τ n = 1 2 2 * -2 2 * -1 =: τ for every j ∈ N \ {0}. Hence, 1 -τ j τ j ≤ 1 -τ τ =: β.
Using that s ≤ 1 and C > 1, this implies that

C 2 5 j (s -t) 2 1-τ j τ j ≤ C 2 5 j (s -t) 2 β . By Young's inequality, ˆBt U γ j dx ≤ (1 -τ j ) ˆBs U γ j + 1 dx + τ j C 2 5 j (s -t) 2 β ˆBt U γ j-1 dx γ j γ j-1 ≤ (1 -τ ) ˆBs U γ j dx + C 2 5 j β (s -t) 2 β ˆBt U γ j-1 dx γ j γ j-1 + |B R |.
By applying Lemma 2.5 with

Z(t) = ˆBt U γ j dx, α 0 = 2 β, and ϑ = 1 -τ ,
we finally obtain (5.9)

ˆBr U γ j dx ≤ C 2 5 j β (R -r) -2 β ˆBR U γ j-1 dx γ j γ j-1 + 1 ,
for some C = C(N, p) > 1.

Step 3: Moser's iteration. We now want to iterate the previous estimate on a sequence of shrinking balls. We fix two radii 0 < r < R ≤ 1, then we consider the sequence

R j = r + R -r 2 j-1 , j ∈ N \ {0},
and we apply (5.9) with R j+1 < R j instead of r < R. Thus we get (5.10)

ˆBR j+1 U γ j dx ≤ C   2 7 j β (R -r) -2 β ˆBR j U γ j-1 dx γ j γ j-1 + 1  
where the constant C > 1 depends on N and p only.

We introduce the notation

Y j = ˆBR j U γ j-1 dx,
thus (5.10) rewrites as

Y j+1 ≤ C 2 7 j β (R -r) -2 β Y γ j γ j-1 j + 1 ≤ C 2 7 j β (R -r) -2 β (Y j + 1) γ j γ j-1 .
Here, we have used again that R ≤ 1. By iterating the previous estimate starting from j = 1, we obtain

Y n+1 ≤ C 2 7 β (R -r) -2 β n-1 j=0 (n-j) γn γ n-j Y 1 + 1 γn γ 0 .
We take the power 1/(γ n ) on both sides:

Y 1 γn n+1 ≤ C 2 7 β (R -r) -2 β n-1 j=0 n-j γ n-j Y 1 + 1 1 γ 0 = C 2 7 β (R -r) -2 β n j=1 j γ j Y 1 + 1 1 γ 0 .
We observe that γ j ∼ 2 j+1 as j goes to ∞. This implies the convergence of the series above and we thus get

U L ∞ (Br) = lim n→∞ ˆBR n+1 U γ n+1 dx 1 γ n+1 ≤ C (R -r) -β ˆBR U p+2 dx + 1 1 p+2
, for some C = C(N, p) > 0 and β = β (N, p) > 0. We also used that γ 0 = p + 2. By recalling the definition of U, we finally obtain (5.11)

∇u L ∞ (Br) ≤ C (R -r) -β ˆBR |∇u| p+2 dx + 1 1 p+2
.

Step 4: L ∞ -L p estimate We fix two concentric balls B r 0 ⊂ B R 0 B with R 0 ≤ 1. Then for every r 0 ≤ t < s ≤ R 0 from (5.11) we have

∇u L ∞ (Bt) ≤ C (s -t) β ˆBs |∇u| p+2 dx 1 p+2 + C (s -t) β ,
where we also used the subadditivity of τ → τ 1/(p+2) . We now observe that

C (s -t) β ˆBs |∇u| p+2 dx 1 p+2 ≤ C (s -t) β ˆBs |∇u| p dx 1 p+2 ∇u 2 p+2 L ∞ (Bs) ≤ 2 p + 2 ∇u L ∞ (Bs) + p p + 2 C (s -t) β p+2 p ˆBs |∇u| p dx 1 p
.

We can apply again Lemma 2.5, this time with the choices

Z(t) = ∇u L ∞ (Bs) , A = p p + 2 C p+2 p ˆBs |∇u| p dx 1 p , α 0 = p + 2 p β .
This yields

∇u L ∞ (Br 0 ) ≤ C   1 (R 0 -r 0 ) β p+2 p ˆBR 0 |∇u| p dx 1 p + 1 (R 0 -r 0 ) β   ,
for every R 0 ≤ 1. This readily implies the desired estimate (5.3) in the homogeneous case. Step 1: a first iterative scheme. This times, we add on both sides of inequality (4.6) the term ˆ|∇η| 2 (|u

x k | -δ k ) p + |u x k | 2 q dx.
The left-hand side then becomes ˆ ∇ (|u

x k | -δ k ) p 2 + |u x k | q η 2 dx
which by Sobolev inequality is greater, up to a constant, than

ˆ(|u x k | -δ k ) 2 * p 2 + |u x k | 2 * q η 2 * dx 2 2 *
.

By summing over k = 1, . . . , N and using Minkowski inequality, we obtain the analogue of (5.4), namely

ˆ N k=1 (|u x k | -δ k ) p + |u x k | 2 q 2 * 2 η 2 * dx 2 2 * ≤ Cq 5 N i,k=1 ˆg i,ε (u x i ) |u x k | 2q+2 |∇η| 2 dx + C q 5 N k=1 ˆ|∇f | |u x k | 2 q+1 η 2 dx + C ˆ|∇η| 2 N k=1 (|u x k | -δ k ) p + |u x k | 2 q dx.
We now introduce the function

U(x) := 1 2 δ max k=1,...,N |u x k (x)|,
where the parameter δ is defined in (2.1). We use that

N k=1 (|u x k | -δ k ) p + |u x k | 2 q ≥ (2 δ U -δ) p + |2 δ U| 2 q ≥ (2 δ) 2 q+p U - 1 2 p + U 2 q ,
and also that for every 1 ≤ i, k ≤ N ,

g i,ε (u x i ) = (p -1) (|u x i | -δ i ) p-2 + + ε ≤ C δ p-2 U p-2 + ε.
This yields

  ˆ U - 1 2 2 * 2 p + U 2 * q η 2 * dx   2 2 * ≤ C q 5 ˆU2 q+p |∇η| 2 dx + C q 5 ε ˆU2 q+2 |∇η| 2 dx + C q 5 ˆ|∇f | U 2 q+1 η 2 dx for a possibly different C = C(N, p, δ) > 1.
With the balls B r ⊂ B t ⊂ B s ⊂ B R and the function η as defined in (5.6), an application of Hölder's inequality leads to

  ˆBt U - 1 2 2 * 2 p + U 2 * q dx   2 2 * ≤ C q 5 (s -t) 2 ˆBs U 2 q+p dx + C q 5 (s -t) 2 ε ˆBs U 2 q+2 dx + C q 5 ∇f L h (B R ) ˆBs U (2 q+1) h dx 1 h
.

(5.12)

From now on, we assume that

(5.13) q ≥ max p -2 h 2 (h -1) , 2 * p 4 h -1 .
This in particular implies that 2 q + 2 ≤ 2 q + p ≤ (2 q + 2) h , then by using Hölder's inequality and taking into account that s ≤ 1, we get

  ˆBt U - 1 2 2 * 2 p + U 2 * q dx   2 2 * ≤ C q 5 (s -t) 2 ˆBs U (2 q+2) h dx 2 q+p (2 q+2) h + C q 5 (s -t) 2 ε ˆBs U (2 q+2) h dx 1 h + C q 5 ∇f L h (B R ) ˆBs U (2 q+2) h dx 2 q+1 (2 q+2) h .
Thanks to the relation on the exponents, this gives (recall that ε < 1 and s ≤ 1)

  ˆBt U - 1 2 2 * 2 p + U 2 * q dx   2 2 * ≤ C q 5 (s -t) 2 1 + ∇f L h (B R ) × ˆBs U (2 q+2) h dx + 1 2 q+p (2 q+2) h . (5.14) We now estimate ˆBs U (2 q+2) h dx = ˆBs∩{U≥1} U (2 q+2) h dx + ˆBs∩{U≤1} U (2 q+2) h dx ≤ ˆBs∩{U≥1} U (2 q+2) h dx + C.
Observe that on the set {U ≥ 1}, we have U ≤ 2 (U -1/2) + . Hence, (5.15)

ˆBs U (2 q+2) h dx ≤ C ˆBs U - 1 2 2 * 2 p + U (2 q+2) h -2 * 2 p dx + C.
We deduce from (5.14) that

  ˆBt U - 1 2 2 * 2 p + U 2 * q dx   2 2 * ≤ C q 5 (s -t) 2 1 + ∇f L h (B R ) ×   ˆBs U - 1 2 2 * 2 p + U (2 q+2) h -2 * 2 p dx + 1   2 q+p (2 q+2) h , (5.16)
for a constant C = C(N, p, h, δ) > 1. We now take q = 2 j+1 -1 for j ≥ j 0 , where j 0 ∈ N is chosen so to ensure condition (5.13). Then we define the sequence of positive exponents

γ j = (2 q + 2) h - 2 * 2 p = 2 j+2 h - 2 * 2 p, j ≥ j 0 ,
and γ j = 2 * q = 2 * (2 j+1 -1), j ≥ j 0 . In order to simplify the notation, we also introduce the measure

d µ := U - 1 2 2 * 2 p + dx.
From (5.16), we get

ˆBt U γ j dµ 2 2 * ≤ C 2 5 j (s -t) 2 1 + ∇f L h (B R ) ˆBs U γ j dµ + 1 2 2 * γ j + 2 * 2 p γ j + 2 * 2 p .
We now observe that 2 h < (2 * p)/2, which implies (5.17)

γ j γ j ≥ 2 * 2 h > 1, j ≥ j 0 .
It follows that

γ j + 2 * 2 p γ j + 2 * 2 p ≤ γ j γ j .
Hence, we obtain

(5.18) ˆBt U γ j dµ 2 2 * ≤ C 2 5 j (s -t) 2 1 + ∇f L h (B R ) ˆBs U γ j dµ + 1 2 2 * γ j γ j
.

Step 2: filling the gaps. Since γ j-1 < γ j < γ j , for every j ≥ j 0 , we obtain by interpolation in Lebesgue spaces, ˆBt

U γ j dµ ≤ ˆBt U γ j-1 dµ τ j γ j γ j-1 ˆBt U γ j dµ (1-τ j ) γ j γ j
, where 0 < τ j < 1 is given by (5.19)

τ j = γ j γ j - 1
γ j γ j γ j γ j-1 - 1
.

We now rely on (5.18) to get ˆBt

U γ j dµ ≤ ˆBt U γ j-1 dµ τ j γ j γ j-1 ×   C 2 5 j (s -t) 2 (1 + ∇f L h (B R ) ) 2 * γ j 2 γj ˆBs U γ j dµ + 1   1-τ j =   C 2 5 j (s -t) 2 (1 + ∇f L h (B R ) ) 2 * γ j (1-τ j ) 2 γj τ j ˆBt U γ j-1 dµ γ j γ j-1   τ j × ˆBs U γ j dµ + 1 1-τ j .
(5.20)

We claim that (5.21)

τ j ≥ τ := 2 * -2 h 4 • 2 * -2 h
for every j ≥ j 0 .

We already know by (5.17) that (γ j /γ j ) ≥ 2 * /(2h ). Moreover, relying on the fact that (2 * p)/2 ≤ 2 j 0 h (this follows from the definition of j 0 ), we also have

2 ≤ γ j γ j-1 ≤ 4, j ≥ j 0 .
By recalling the definition (5.19) of τ j , we get

τ j = ζ γj γ j , γ j γ j-1 , where ζ(x, y) = x -1 x y -1 . Observe that on [2 * /(2 h ), +∞) × [2, 4], the function x → ζ(x, y) is increasing, while y → ζ(x, y) is decreasing. Thus we get τ j ≥ ζ 2 * 2 h , 4 ,
which is exactly claim (5.21). We deduce from (5.21) and (5.17) that

2 * γ j (1 -τ j ) 2 γj τ j ≤ 1 -τ τ h =: β.
In particular, we have

C 2 5 j (s -t) 2 (1 + ∇f L h (B R ) ) 2 * γ j (1-τ j ) 2 γj τ j ≤ C 2 5 j (s -t) 2 (1 + ∇f L h (B R ) ) β ,
since the quantity inside the parenthesis is larger than 1 (here, we use again that s ≤ 1). In view of (5.20), this implies ˆBt Step 3: Moser's iteration. Estimate (5.22) is the analogue of (5.9), except that the Lebesgue measure dx is now replaced by the measure dµ, and the index j is assumed to be larger than some j 0 , instead of j ≥ 0 as in (5.9). Following the same iteration argument, we are led to (5.23)

U γ j dµ ≤ C 2 5 j (s -t) 2 (1 + ∇f L h (B R ) ) β ˆBt U γ j-1 dµ γ j γ j-1 τ j × ˆBs U γ j dµ + 1 1-τ j . By Young's inequality, ˆBt U γ j dµ ≤ (1 -τ j ) ˆBs U γ j dµ + 1 + τ j C 2 5 j (s -t) 2 (1 + ∇f L h (B R ) ) β ˆBt U γ j-1 dµ γ j γ j-1 ≤ (1 -τ ) ˆBs U γ j dµ + C 2 5 j β (s -t) 2 β (1 + ∇f L h (B R ) ) β ˆBR U γ j-1 dµ γ j γ j-1 + 1,
U L ∞ (Br, dµ) ≤ C 1 + ∇f L h (B R ) R -r β ˆBR U γ j 0 dµ + 1 1 γ j 0 ,
for some C = C(N, p, h, δ) > 1, β = β (N, p, h) > 0.

Step 4: L ∞ -L p estimate. We now want to replace the norm L γ j 0 of U in the right-hand side of (5.23) by its norm L p . Let q 1 := 2 j 1 +1 -1 where j 1 := min j ≥ j 0 : j + 1 ≥ log 2 1 + γ j 0 2 * .

Then γ j 0 ≤ 2 * q 1 and thus, by using that U γ j 0 ≤ 2 2 * q 1 -γ j 0 U 2 * q 1 , whenever U ≥ 1 2 , we have (5.24) U L γ j 0 (B R , dµ) ≤ C U 2 * q 1 γ j 0 L 2 * q 1 (B R , dµ) .

We rely on (5.14) with q = q 1 to get for every 0 < r < t < s < R

(5.25) U 2 q 1 L 2 * q 1 (Bt, dµ) ≤ C (s -t) 2 1 + ∇f L h (B R ) U 2 q 1 +p L 2 (q 1 +1) h (Bs)

+ 1 ,
for some new constant C = C(N, p, h, δ) > 1.

Since j ≥ j 0 , we have p < (2 q 1 + 2) h < 2 * /2 (2 q 1 + p), and thus, by interpolation in Lebesgue spaces (5.26) U L 2 (q 1 +1) h (Bs) ≤ U θ L 2 * q 1 + 2 * 2 p (Bs)

U 1-θ L p (Bs) ,
where θ ∈ (0, 1) is determined as usual by scale invariance. As in the proof of (5.15), we have

U L 2 * q 1 + 2 * 2 p (Bs)
≤ C U 2 q 1 2 q 1 +p L 2 * q 1 (Bs, dµ) + C.

Inserting this last estimate into (5.26), we obtain U 2 q 1 +p L 2 (q 1 +1) h (Bs) ≤ C U 2 q 1 θ L 2 * q 1 (Bs, dµ) U

(1-θ) (2 q 1 +p) L p (Bs)

+ C U (1-θ) (2 q 1 +p) L p (Bs)
, up to changing the constant C = C(N, p, h, δ) > 1. In view of (5.25), this gives U 2 q 1 L 2 * q 1 (Bt, dµ) ≤ C (s -t) 2 1 + ∇f L h (B R ) U 2 q 1 θ L 2 * q 1 (Bs, dµ) U

(1-θ) (2 q 1 +p) L p (Bs)

+ U

(1-θ) (2 q 1 +p) L p (Bs) + 1 .

By Young's inequality, we get U 2 q 1 L 2 * q 1 (Bt, dµ) ≤ θ U 2 q 1 L 2 * q 1 (Bs, dµ) + (1 -θ)

C (s -t) 2 (1 + ∇f L h (B R ) ) 1 1-θ U (2 q 1 +p) L p (Bs) + C (s -t) 2 1 + ∇f L h (B R )
U

(1-θ) (2 q 1 +p) L p (Bs) + 1 .

By Lemma 2.5, this implies

U 2 q 1 L 2 * q 1 (Br, dµ) ≤ C 1 (R -r) 2 (1 + ∇f L h (B R 0 ) ) 1 1-θ U (2 q 1 +p) L p (B R ) + 1 .
Coming back to (5.23) and taking into account (5.24), we obtain

U L ∞ (Br 0 , dµ) ≤ C 1 + ∇f L h (B R 0 ) R 0 -r 0 σ 2 U σ 1 L p (B R 0 ) + 1 ,
where C = C(N, p, h, δ) > 1 and σ i = σ i (N, p, h) > 0, for i = 1, 2. By definition of U, we have

|∇u| ≤ 2 δ √ N U ≤ 2 δ √ N |∇u|.
Since U L ∞ (Br 0 , dµ) + 1 ≥ U L ∞ (Br 0 ) , it follows that

∇u L ∞ (Br 0 ) ≤ C 1 + ∇f L h (B R 0 ) R 0 -r 0 σ 2 ∇u σ 1 L p (B R 0 ) + 1 .
This completes the proof.

where

G ε (v; B) = N i=1 ˆB g i,ε (v x i ) dx + ˆB G ε (x, v) dx
and U ε = U * ρ ε . Moreover, by [8, Remark 7.6] we have u ε ∈ L ∞ (B), with a bound on the L ∞ norm uniform in ε > 0. In order to simplify the notation, we simply write as usual u and f instead of u ε and f ε . The Euler equation is now N i=1 ˆg i,ε (u x i ) ϕ x i dx + ˆf (x, u) ϕ dx = 0, ϕ ∈ W 1,p 0 (B).

When we differentiate the Euler equation with respect to some direction x j , we obtain

N i=1 ˆg i,ε (u x i ) u x i x j ψ x i dx + 1 p -1 ˆ f x j (x, u) + f ξ (x, u) u x j ψ dx = 0, ψ ∈ W 1,p 0 (B).
We can then repeat the proof of Proposition 5.1 with this additional term f ξ (x, u)u x j which leads to the following analogue of (5.12):

  ˆBt U - 1 2 2 * 2 p + U 2 * q dx   2 2 * ≤ C q 5 (s -t) 2 ˆBs U 2 q+p dx + C q 5 (s -t) 2 ε ˆBs U 2 q+2 dx + C q 5 ∇ x f L h ˆBs U (2 q+1) h dx 1 h + C q 5 f ξ L h ˆBs U (2 q+2) h dx 1 h
.

Using again Hölder's inequality for the first three terms, we obtain inequality (5.14) where ∇f now represents the full gradient of f with respect to both x and ξ. The rest of the proof is the same and leads to a uniform Lipschitz estimate, as desired.

  2 u x i x i = 0, whose local weak solutions are local minimizers of the functional(1.3) I(u, Ω ) = 1 p ˆΩ |∇u| p dx, Ω Ω, u ∈ W 1,p loc (Ω ).

  The paper has been partially written during a visit of P. B. & L. B. to Napoli and of C. L. to Ferrara. Both visits have been funded by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) through the project "Regolarità per operatori degeneri con crescite generali ". A further visit of P. B. to Ferrara in April 2017 has been the occasion to finalize the work. Hosting institutions are gratefully acknowledged. The last three authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Proposition 4 . 1 (

 41 ) = t m , for t ≥ 0, with 1 ≤ s ≤ m. Staircase to the full Caccioppoli). Let p ≥ 2 and let η ∈ C ∞ 0 (B), then for every k, j = 1, . . . , N and 1

5. 2 .

 2 Proof of Proposition 5.1: the non-homogeneous case. We follow step by step the proof of the homogeneous case and we only indicate the main changes, which essentially occur in Step 1 and Step 2.

  where C = C(N, p, h, δ) > 1 as usual. By applying again Lemma 2.5, this times with the choicesZ(t) = ˆBt U γ j dµ, α 0 = 2 β, and ϑ = 1 -τ ,we finally obtain(5.22)ˆBr U γ j dµ ≤ C 2 5 j β (R -r) 2 β (1 + ∇f L h (B R ) ) β ˆBR U γ j-1 dµ

This test function is not really admissible, since it is not compactly supported. Actually, to make it admissible we have to multiply it by a cut-off function. However, this gives unessential modifications, we prefer to avoid it in order to neatly present the idea of the proof.

Appendix A. Lipschitz regularity with a nonlinear lower order term

In this section, we consider the functional

The lower order term f u of the functional F δ is thus replaced by a more general term G(x, u). We assume that for almost every x ∈ Ω, the map

We denote f (x, ξ) := G ξ (x, ξ) and we assume that f ∈ W 1,h loc (Ω × R), for some h > N/2. Finally, we assume that G(x, ξ) satisfies the inequality

where 1 < p ≤ γ < p * and a, b are two non-negative functions belonging respectively to L s loc (Ω) and L σ loc (Ω) with s > N/p and σ > p * /(p * -γ). Under assumption (A.1), all the local minimizers of G δ are locally bounded, see [8, Theorem 7.5] and moreover, for every such minimizer u, for every

where M depends on u W 1,p (B R 0 ) , r 0 , R 0 , b L σ (R 0 ) , and a L s (B R 0 ) .

Then we have:

Theorem A.1. Let p ≥ 2 and let u ∈ W 1,p loc (Ω) be a local minimizer of the functional G δ . Then u is locally Lipschitz in Ω.

Proof. We only explain the main differences with respect to the proof of Theorem 1.1. Since G is convex with respect to the second variable, the functional G is still convex. This implies that Lemma 2.3 remains true with the same proof. We then introduce the approximation of G:

where ρ ε is the same regularization kernel as before, while ρ ε is a regularization kernel on R. Given a local minimizer U ∈ W 1,p loc (Ω) and a ball B ⊂ 2 B Ω, there exists a unique C 2 solution u ε to the regularized problem min G ε (v; B) : v -U ε ∈ W 1,p 0 (B) ,