J Ér

Ôme V Étois

Jérôme Vétois
email: vetois@unice.fr

EXISTENCE AND REGULARITY FOR CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

We establish existence and regularity results for doubly critical anisotropic equations in domains of the Euclidean space. In particular, we answer a question posed by when the maximum of the anisotropic configuration coincides with the critical Sobolev exponent.

Introduction

In this paper, we investigate existence and regularity for doubly critical anisotropic equations. In dimension n ≥ 2, we provide ourselves with an anisotropic configuration -→ p = (p 1 , . . . , p n ) with p i > 1 for all i = 1, . . . , n. We let D 1, - → p (Ω) be the anisotropic Sobolev space defined as the completion of the vector space of all smooth functions with compact support in Ω with respect to the norm u D 1, - → p (Ω) = n i=1 ∂u/∂x i L p i (Ω) . We are concerned with the following anisotropic problem of critical growth

-∆ - → p u = λ |u| p * -2 u in Ω , u ∈ D 1, - → p (Ω) , (1.1)
on domains Ω in the Euclidean space R n , where λ is a positive real number, p * is the critical Sobolev exponent (see (1.3) below), and ∆ - → p is the anisotropic Laplace operator defined by

∆ - → p u = n i=1 ∂ ∂x i ∇ p i x i u , (1.2)
where ∇ p i x i u = |∂u/∂x i | p i -2 ∂u/∂x i for all i = 1, . . . , n. As one can check, ∆ - → p involves directional derivatives with distinct weights. Anisotropic operators appear in several places in the literature. Recent references can be found in physics [START_REF] Antontsev | Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics[END_REF][START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF], in biology [START_REF] Bendahmane | On some anisotropic reaction-diffusion systems with L 1 -data modeling the propagation of an epidemic disease[END_REF], and in image processing [START_REF] Weickert | Anisotropic diffusion in image processing[END_REF]. We consider in this paper the doubly critical situation p + = p * , where p + = max (p 1 , . . . , p n ) is the maximum value of the anisotropic configuration and p * is the critical Sobolev exponent for the embeddings of the anisotropic Sobolev space D 1, - → p (Ω) into Lebesgue spaces. In this setting, not only the nonlinearity has critical growth, but the operator itself has critical growth in particular directions of the Euclidean space. As a remark, the notion of critical direction is a pure anisotropic notion which does not exist when dealing with the Laplace operator or the p-Laplace operator. Given i = 1, . . . , n, the i-th direction is said to be critical if p i = p * , resp. subcritical if p i < p * . Critical directions induce a failure in the rescaling invariance rule associated with (1.1).

Given an anisotropic configuration -→ p satisfying n i=1 1/p i > 1 and p j ≤ n/ n i=1 1 p i -1 for all j = 1, . . . , n, the critical Sobolev exponent is equal to

p * = n n i=1 1 p i -1 . (1.3)
In this paper, we consider weak solutions of problem (1.1). We say that a function u in D 1, - → p (Ω) is a weak solution of problem (1.1) if there holds

n i=1 Ω ∂u ∂x i p i -2 ∂u ∂x i ∂ϕ ∂x i dx = Ω |u| p * -2 uϕdx
for all smooth functions ϕ with compact support in Ω.

In this paper, we prove an existence result and a regularity result for problem (1.1). The regularity result, stated in Theorem 1.2 below, is established on arbitrary domains (bounded or not), and is motivated in particular by a question posed by Fragalà-Gazzola-Kawohl [24, Section 8.3, Problem 1]. The existence result, stated in Theorem 1.1 below, is established on cylindric domains. Problem (1.1) on cylindric domains is involved in the description of the asymptotic behavior of Palais-Smale sequences for critical anisotropic problems (see Vétois [START_REF]The blow-up of critical anistropic equations with critical directions[END_REF]). The rescaling phenomenon is described in Section 3. Our existence result states as follows.

Theorem 1.1. Let n ≥ 3, 1 ≤ n + < n, and -→ p = (p 1 , . . . , p n ), and assume that n i=1 1/p i > 1,

p + = p * , p n-n + +1 = • • • = p n = p + ,
and p i < p + for all i ≤ n -n + . Let V be a nonempty, bounded, open subset of R n + , and assume that Ω = R n-n + × V . Then there exists a positive real number λ such that problem (1.1) admits at least one nonnegative, nontrivial solution.

Theorem 1.1 is concerned with cylindric domains. Theorem 1.2 below holds true for arbitrary domains Ω, including Ω bounded. This result, which answers the question of the regularity associated to (1.1), is stated as follows. The boundedness of nonnegative weak solutions of problem (1.1) was established in case p + < p * by Fragalà-Gazzola-Kawohl [START_REF] Fragalà | Existence and nonexistence results for anisotropic quasilinear elliptic equations[END_REF]. It was suggested in [START_REF] Fragalà | Existence and nonexistence results for anisotropic quasilinear elliptic equations[END_REF] that the result should remain true in case p + ≥ p * for solutions of the problem

-∆ - → p u = λu p + -1 in Ω , u ∈ D 1, - → p (Ω) ∩ L p + (Ω) . (1.4)
Theorem 1.2 answers positively to this question in case p + = p * . On the other hand, we point toward a negative answer when p + > p * . More precisely, we prove (by using Proposition 2.1, see Section 2) that for particular anisotropic configurations -→ p satisfying p + > p * , for instance when

p 1 = • • • = p n -= 2 and p n -+1 = • • • = p n = p + with p + > 2 * , 2 * = 2n -/ (n --2)
, and 2 < n -< n, if we assume the existence of nonnegative, unbounded solutions of the isotropic, supercritical problem

-∆u = u p + -1 in Ω ′ , u ∈ D 1,2 (Ω ′ ) ∩ L p + (Ω ′ ) ,
for some domain Ω ′ in R n -, where ∆ = div (∇u) is the classical Laplace operator, then the anisotropic problem (1.4) with Ω = Ω ′ × Ω ′′ admits nonnegative, unbounded solutions for all domains Ω ′′ in R n-n -, including Ω ′′ bounded. As is well-known, problems with supercritical growth may admit unbounded solutions (see, for instance, Benguria-Dolbeault-Esteban [START_REF] Benguria | Classification of the solutions of semilinear elliptic problems in a ball[END_REF], Farina [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R n[END_REF], and also Fragalà-Gazzola-Kawohl [START_REF] Fragalà | Existence and nonexistence results for anisotropic quasilinear elliptic equations[END_REF]).

In case p + < p * , namely when all directions are subcritical, anisotropic equations with critical nonlinearities have been investigated by Alves-El Hamidi [START_REF] Alves | Existence of solution for a anisotropic equation with critical exponent[END_REF], El Hamidi-Rakotoson [START_REF] Hamidi | On a perturbed anisotropic equation with a critical exponent[END_REF][START_REF]Extremal functions for the anisotropic Sobolev inequalities[END_REF], El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF], Fragalà-Gazzola-Kawohl [START_REF] Fragalà | Existence and nonexistence results for anisotropic quasilinear elliptic equations[END_REF], Fragalà-Gazzola-Lieberman [START_REF] Fragalà | Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains[END_REF], and Vétois [START_REF]Asymptotic stability, convexity, and Lipschitz regularity of domains in the anisotropic regime[END_REF]. Other recent references on anisotropic problems like (1.1) are Antontsev-Shmarev [START_REF] Antontsev | Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions[END_REF][START_REF]Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions[END_REF], Bendahmane-Karlsen [START_REF] Bendahmane | Renormalized solutions of an anisotropic reaction-diffusion-advection system with L 1 data[END_REF][START_REF]Nonlinear anisotropic elliptic and parabolic equations in R N with advection and lower order terms and locally integrable data[END_REF], Bendahmane-Langlais-Saad [START_REF] Bendahmane | On some anisotropic reaction-diffusion systems with L 1 -data modeling the propagation of an epidemic disease[END_REF], Cianchi [START_REF] Cianchi | Symmetrization in anisotropic elliptic problems[END_REF], D'Ambrosio [START_REF] Ambrosio | Liouville theorems for anisotropic quasilinear inequalities[END_REF], Di Castro [START_REF] Castro | Existence and regularity results for anisotropic elliptic problems[END_REF], Di Castro-Montefusco [START_REF] Castro | Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations[END_REF], García-Melián-Rossi-Sabina de Lis [START_REF] García-Melián | Large solutions to an anisotropic quasilinear elliptic problem[END_REF], Li [START_REF] Li | Anisotropic elliptic equations in L m[END_REF], Lieberman [START_REF] Lieberman | Gradient estimates for a new class of degenerate elliptic and parabolic equations[END_REF][START_REF]Gradient estimates for anisotropic elliptic equations[END_REF], Mihȃilescu-Pucci-Rȃdulescu [START_REF] Mihȃilescu | Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent[END_REF], Mihȃilescu-Rȃdulescu-Tersian [START_REF] Mihȃilescu | Eigenvalue problems for anisotropic discrete boundary value problem[END_REF], Namlyeyeva-Shishkov-Skrypnik [START_REF] Namlyeyeva | Isolated singularities of solutions of quasilinear anisotropic elliptic equations[END_REF], Skrypnik [START_REF] Skrypnik | Removability of an isolated singularity for anisotropic elliptic equations with absorption[END_REF], Tersenov-Tersenov [START_REF] Tersenov | The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations[END_REF], and Vétois [START_REF] Vétois | A priori estimates for solutions of anisotropic elliptic equations[END_REF][START_REF]The blow-up of critical anistropic equations with critical directions[END_REF][START_REF]Strong maximum principles for anisotropic elliptic and parabolic equations[END_REF].

In the isotropic configuration where p i = p for all i = 1, . . . , n, there holds p < p * and all directions are subcritical. In this particular situation, the operator (1.2) is comparable, though slightly different, to the p-Laplace operator ∆ p = div |∇u| p-2 ∇u . Possible references on critical p-Laplace equations are Alves-Ding [START_REF] Alves | Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity[END_REF], Arioli-Gazzola [START_REF] Arioli | Some results on p-Laplace equations with a critical growth term[END_REF], Demengel-Hebey [START_REF] Demengel | On some nonlinear equations involving the p-Laplacian with critical Sobolev growth[END_REF][START_REF]On some nonlinear equations involving the p-Laplacian with critical Sobolev growth and perturbation terms[END_REF], Filippucci-Pucci-Robert [START_REF] Filippucci | On a p-Laplace equation with multiple critical nonlinearities[END_REF], Gazzola [START_REF] Gazzola | Critical growth quasilinear elliptic problems with shifting subcritical perturbation[END_REF], and Guedda-Veron [START_REF] Guedda | Quasilinear elliptic equations involving critical Sobolev exponents[END_REF]. Needless to say, the above list does not pretend to exhaustivity.

We illustrate our results with examples in Section 2, we prove Theorem 1.1 in Section 3, and we prove Theorem 1.2 in Section 4.

Examples of solutions

In this section, we are concerned with the situation where the anisotropic configuration -→ p consists in two distinct exponents p -and p + . In other words, we assume that there exist two indices n -≥ 2 and n + ≥ 1 such that n = n -+ n + , p 1 = • • • = p n -= p -, and p n -+1 = • • • = p n = p + . Proposition 2.1 below is the basic tool in our construction. It relies on a direct computation. Proposition 2.1. Let n -≥ 2, n + ≥ 1, n = n -+ n + , and -→ p = (p 1 , . . . , p n ), and assume that

p 1 = • • • = p n -= p -and p n -+1 = • • • = p n = p + . Let λ be a positive real number. Let Ω 1 be a nonempty open subset of R n -and Ω 2 be a nonempty open subset of R n + . Let v be a solution of the problem      - n - i=1 ∂ ∂x i ∂v ∂x i p --2 ∂v ∂x i = |v| p + -2 v in Ω 1 , v ∈ D 1,p -(Ω 1 ) ∩ L p + (Ω 1 ) , (2.1)
and let w be a solution of the problem

     - n + i=1 ∂ ∂x i ∂w ∂x i p + -2 ∂w ∂x i = |w| p + -2 w -|w| p --2 w in Ω 2 , w ∈ D 1,p + (Ω 2 ) ∩ L p -(Ω 2 ) .
(2.2)

Then the function u defined on

Ω 1 × λ -1 p + Ω 2 by u (x 1 , . . . , x n ) = λ -1 p + -p -v x 1 , . . . , x n -w λ 1 p + x n -+1 , . . . , λ 1 p + x n (2.3) is a solution of the problem    -∆ - → p u = λ |u| p + -2 u in Ω 1 × λ -1 p + Ω 2 , u ∈ D 1, - → p Ω 1 × λ -1 p + Ω 2 ∩ L p + Ω 1 × λ -1 p + Ω 2 , (2.4)
where ∆ - → p is as in (1.2).

Proof. A direct computation provides the result.

If p + = p * , then a solution of equation (2.1) is given by

V n -,p -x 1 , . . . , x n -= C n -,p - 1 1 + n - i=1 |x i | p - p --1 n --p - p - , (2.5)
where

C n -,p -= n -(n --p -) p --1 (p --1) p --1 n --p - p 2 - .
On the other hand, we search for solutions of equation (2.2) of the form

w x n -+1 , . . . , x n = W (r) with r = n i=n -+1 |x i | p + p + -1 p + -1 p + .
As one can check, equation (2.2) then rewrites as

-r 1-n + r n + -1 |W ′ | p + -2 W ′ ′ = |W| p + -2 W -|W| p --2 W in R + . (2.6)
In case n + = 1, the unique nonnegative, nontrivial C 1 -solution of (2.6) is given by

W (r) = F -1 (F (W 0 ) -r) if r < F (W 0 ) , 0 if r ≥ F (W 0 ) ,
where

W 0 = p + p - 1 p + -p - and F (t) = p + -1 p + 1 p + t 0 s p - p - - s p + p + -1 p + ds .
In particular, there hold W (0) = W 0 , W ′ (0) = 0, W > 0 and W ′ < 0 in (0, F (W 0 )), and W = 0 in [F (W 0 ) , +∞). In case n + ≥ 2, by Franchi-Lanconelli-Serrin [START_REF] Franchi | Existence and uniqueness of nonnegative solutions of quasilinear equations in R n[END_REF], we get that equation (2.6) admits at least one nonnegative C 1 -solution which satisfies W ′ (0) = 0, W > 0 and W ′ < 0 in (0, R), and W = 0 in [R, +∞) for some positive real number R. Summarizing, we can state the following corollary of Proposition 2.1.

Corollary 2.1. Let n -≥ 2, n + ≥ 1, n = n -+ n +
, and -→ p = (p 1 , . . . , p n ), and assume that

p 1 = • • • = p n -= p -, p n -+1 = • • • = p n = p +
, and p + = p * . For any point a = (a 1 , . . . , a n ) in R n and for any positive real numbers µ and λ, there exists a nonnegative solution U a,µ,λ in

D 1, - → p (R n ) ∩ C 1 (R n ) of equation (2.4) of the form U a,µ,λ (x 1 , . . . , x n ) = µ -1 λ -1 p + -p -U µ p --p + p - (x 1 -a 1 ) , . . . , µ p --p + p - x n --a n -, λ 1 p + x n -+1 -a n -+1 , . . . , λ 1 p + (x n -a n ) ,
where

U (x 1 , . . . , x n ) = V n -,p -x 1 , . . . , x n -W   n i=n -+1 |x i | p + p + -1 p + -1 p +   ,
where V n -,p -is as in (2.5) and where W is such that W > 0 and W ′ < 0 in (0, R), and W = 0 in [R, +∞) for some positive real number R.

Since the function W has compact support, Corollary 2.1 provides a class of solutions of problem (1.1) on cylindric domains Ω = R n -× V for all nonempty, open subsets V of R n + . These solutions illustrate the general existence result stated in Theorem 1.1 in the particular case where the anisotropic configuration -→ p consists in two distinct exponents p -and p + .

In the supercritical case p + > p * , suppose there exists a nonnegative, unbounded solution of problem (2.1) for some domain Ω 1 in R n -. Then we easily get with Proposition 2.1 that problem (1.4) with Ω = Ω 1 × Ω 2 admits nonnegative, unbounded solutions for all domains Ω 2 in R n + , including Ω 2 bounded. Indeed, since the above function W has compact support, by rescaling W, we get a nonnegative solution of the problem (2.2) on the domain Ω 2 . Then Proposition 2.1 provides the existence of a nonnegative, unbounded solution of the form (2.3) of the problem (1.4) with Ω = Ω 1 × Ω 2 .

The existence result

This section is devoted to the proof of Theorem 1.1. We let n ≥ 3 and -→ p = (p 1 , . . . , p n ). We assume that n i=1 1/p i > 1, p + = p * , and that there exists an index n + such that p n-n + +1 = • • • = p n = p + , and p i < p + for all i ≤ n -n + . Moreover, we assume that Ω = R n-n + × V , where V is a nonempty, bounded, open subset of R n + . Without loss of generality, we may assume that the point 0 belongs to V .

The proof of Theorem 1.1 is based on concentration-compactness arguments. Let us first set some notations. For any function u in D 1, - → p (R n ) and any subset D of R n , we let the energy E (u, D) of u on D be defined by

E (u, D) = D u p + dx . (3.1)
For any positive real number µ and any point a = (a 1 , . . . , a n ) in R n , we define the affine transformation τ

- → p µ,a : R n → R n by τ - → p µ,a (x 1 , . . . , x n ) = µ p 1 -p + p 1 (x 1 -a 1 ) , . . . , µ pn-p + pn (x n -a n ) . (3.2)
As is easily checked, (3.2) provides a general rescaling invariance rule associated with equation (1.1). Moreover for any subset

D of R n , we get E (u, D) = E µu • τ - → p µ,a -1 , τ - → p µ,a (D) , where τ - → p µ,a -1 (x 1 , . . . , x n ) = µ p + -p 1 p 1 x 1 + a 1 , . . . , µ p + -pn pn x n + a n .
Of importance in our critical setting is that the set D is only rescaled with respect to noncritical directions. Therefore, we observe a concentration phenomenon on affine subspaces of R n spanned by critical directions. Figure 1 below illustrates the rescaling in case D is a threedimensional ball, the first two directions being noncritical, the third one being critical. In case of the p-Laplace operator, the ball would have been rescaled to the whole euclidean space.

1 p i Ω ∂u α ∂x i p i dx = inf u∈D 1, - → p (Ω) Ω |u| p + dx=1 n i=1 1 p i Ω ∂u ∂x i p i dx . (3.3)
Taking the absolute value, we may assume that for any α, the function u α is nonnegative.

Clearly, the sequence (u α ) α is bounded in D 1, - → p (Ω).

Step 3.1 below is the first step in the proof of Theorem 1.1. We say that a sequence

(v α ) α in D 1, - → p (Ω) is Palais-Smale for the functional I λ defined in (3.4) if there hold |I λ (v α )| ≤ C
for some positive constant C independent of α, and

DI λ (v α ) D 1, - → p (Ω) ′ → 0 as α → +∞.
Step 3.1. Up to a subsequence, (u α ) α is a Palais-Smale sequence for the functional

I λ (u) = n i=1 1 p i Ω ∂u ∂x i p i dx - λ p + Ω |u| p + dx , (3.4)
where

λ = lim α→+∞ n i=1 Ω ∂u α ∂x i p i dx . (3.5)
Proof. It easily follows from (3.3) that there holds |I λ (u α )| ≤ C for some positive contant C independent of α. We then prove that for any bounded sequence (ϕ α ) α in D 1, - → p (Ω), there holds DI λ (u α ) .ϕ α → 0 as α → +∞. By (3.3), we get that there exists a sequence (ε α ) α of positive real numbers converging to 0 such that for any real number t, there holds

n i=1 1 p i Ω ∂u α ∂x i p i dx -ε α ≤ n i=1 1 p i Ω ∂ ∂x i   u α + tϕ α Ω |u α + tϕ α | p + dx 1 p +   p i dx = n i=1 1 p i Ω |u α + tϕ α | p + dx - p i p + Ω ∂u α ∂x i + t ∂ϕ α ∂x i p i dx . (3.6)
As is easily checked, there exists a positive real number C such that for any i = 1, . . . , n and for any real numbers x and y, there holds

|x + y| p i -|x| p i -p i |x| p i -2 xy ≤ C |y| p i if p i ≤ 2 |y| 2 |x| p i -2 + |y| p i -2 if p i > 2. (3.7)
Since (u α ) α and (ϕ α ) α are bounded in D 1, - → p (Ω), by (3.7) and Hölder's inequality, we get

Ω ∂u α ∂x i + t ∂ϕ α ∂x i p i dx - Ω ∂u α ∂x i p i dx -p i t Ω ∂u α ∂x i p i -2 ∂u α ∂x i ∂ϕ α ∂x i dx ≤ C          t p i Ω ∂ϕ α ∂x i p i dx if p i ≤ 2 t 2 Ω ∂u α ∂x i p i dx 2 p i Ω ∂ϕ α ∂x i p i dx p i -2 p i + t p i Ω ∂ϕ α ∂x i p i dx if p i > 2. ≤ C ′ t p i if p i ≤ 2 t 2 1 + t p i -2 if p i > 2. ( 3.8)
for all i = 1, . . . , n, and

Ω |u α + tϕ α | p + dx - Ω u p + α dx -p + t Ω u p + -1 α ϕ α dx ≤ C          t p + Ω |ϕ α | p + dx if p + ≤ 2 t 2 Ω |u α | p + dx 2 p + Ω |ϕ α | p + dx p + -2 p + + t p + Ω |ϕ α | p + dx if p + > 2. ≤ C ′ t p + if p + ≤ 2 t 2 1 + t p + -2 if p + > 2. (3.9)
for some positive constants C and C ′ independent of α and t. By (3.6), (3.8), (3.9), we get

-ε α ≤ t n i=1 Ω ∂u α ∂x i p i -2 ∂u α ∂x i ∂ϕ α ∂x i dx - n i=1 Ω ∂u α ∂x i p i dx Ω u p + -1 α ϕ α dx + o (t) ≤ t DI λ (u α ) .ϕ α + λ - n i=1 Ω ∂u α ∂x i p i dx Ω u p + -1 α ϕ α dx + o (t)
as t → 0 uniformly with respect to α, where λ is as in (3.5). Passing to the limit as α → +∞, we get 0 ≤ lim sup α→+∞ (tDI λ (u α ) .ϕ α ) + o (t) as t → 0. Since the real number t takes either positive or negative values, it follows that DI λ (u α ) .ϕ α → 0 as α → +∞. Since this holds true for all bounded sequences (ϕ α ) α in D 1, - → p (Ω), we get DI λ (u α ) D 1, - → p (Ω) ′ → 0 as α → +∞. This ends the proof of Step 3.1.

Now, for any α, we define the concentration function

Q α : R + → R + by Q α (s) = max y∈Ω E u α , P - → p y (s) ,
where the energy functional E is as in (3.1) and

P - → p y (s) = (x 1 , . . . , x n ) ∈ Ω; |x i -y i | < s p + -p i p i ∀i ∈ {1, . . . , n -n + } (3.10)
for all positive real number s and for all point y = (y 1 , . . . , y n ) in Ω. By the continuity of the functions Q α and by (3.3), given a real number δ 0 in (0, 1), we get the existence of a sequence (µ α ) α of positive real numbers such that there holds Q α (µ α ) = δ 0 for all α. We let x α be a point in Ω for which Q α (µ α ) is reached, so that there holds

max y∈Ω E u α , P - → p y (µ α ) = E u α , P - → p xα (µ α ) = δ 0 (3.11)
for all α. By definition of P -→ p xα (µ α ), see (3.10), we may assume that the n + last coordinates of the point x α are equal to 0. For any α, we then define the function u α by As well as (u α ) α , we get that ( u α ) α is a Palais-Smale sequence for the functional I λ defined in (3.4). Moreover, there holds u α D 1, - → p (Ω) = u α D 1, - → p (Ω) for all α. In particular, the sequence ( u α ) α is bounded in D 1, - → p (Ω). Passing if necessary to a subsequence, we may assume that ( u α ) α converges weakly to a nonnegative function u ∞ in D 1, - → p (Ω) and that ( u α ) α converges to u ∞ almost everywhere in Ω. The second step in the proof of Theorem 1.1 is as follows.

u α = µ α u α • τ - → p µα,xα
Step 3.2. If the constant δ 0 is small enough, then ( u α ) α converges, up to a subsequence, to

u ∞ in L p + loc (R n ). Proof.
We fix a positive real number R, and we let B 0 (R) be the (n -n + )-dimensional ball of center 0 and radius R. We show that the sequence ( u α ) α converges to u ∞ in L p + (B 0 (R)). For any α, we let v α = u α -u ∞ . By Banach-Alaoglu theorem, since the sequence (v α ) α is bounded in D 1, - → p (Ω) and since Ω = R n-n + × V , where V is bounded, passing if necessary to a subsequence, we may assume that there exist nonnegative, finite measures µ and ν 1 , . . . , ν n on B 0 (2R) × R n + such that |v α | p + ⇀ µ and |∂v α /∂x i | p i ⇀ ν i as α → +∞ in the sense of measures on B 0 (2R) × R n + , for all i = 1, . . . , n. Moreover, the supports of the measures µ and ν 1 , . . . , ν n are included in B 0 (2R) × V . Now, we borrow some ideas in Lions [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The limit case. II[END_REF] with the tricky difference here that the concentration holds on n + -dimensional affine subspaces of R n . Since p + = p * , by the anisotropic Sobolev inequality in Troisi [START_REF] Troisi | Teoremi di inclusione per spazi di Sobolev non isotropi[END_REF], there exists a positive constant Λ = Λ ( -→ p ) such that for any α and any smooth function ϕ with compact support in B 0 (2R) × R n + , there holds

Ω |v α ϕ| p + dx ≤ Λ n i=1 Ω ∂ (v α ϕ) ∂x i p i dx p + np i ≤ Λ n i=1 Ω v α ∂ϕ ∂x i p i dx 1 p i + Ω ∂v α ∂x i ϕ p i dx 1 p i p + n . (3.12)
For i = 1, . . . , n-n + , by the compact embeddings in Rákosník [START_REF] Rákosník | Some remarks to anisotropic Sobolev spaces. I[END_REF], we get that (v α ) α converges to 0 in L p i (Supp ϕ). Passing to the limit as α → +∞ into (3.12) gives

B 0 (R)×V |ϕ| p + dµ ≤ Λ n-n + i=1 B 0 (R)×V |ϕ| p i dν i p + np i × n i=n-n + +1 B 0 (R)×V ∂ϕ ∂x i p + dµ 1 p + + B 0 (R)×V |ϕ| p + dν i 1 p + p + n
.

By an easy density argument, it follows that for any bounded measurable function ϕ on B 0 (R) × V which does not depend on the variables x n-n + +1 , . . . , x n , there holds

B 0 (R)×V |ϕ| p + dµ ≤ Λ n i=1 B 0 (R)×V |ϕ| p i dν i p + np i . (3.13)
In particular, for any Borelian set A in B 0 (R), taking ϕ = 1 A×V , we get

µ A × V ≤ Λ n i=1 ν i A × V p + np i . (3.14) Letting ν = n i=1 ν i , since n i=1 1 p i = n+p + p + , it follows that µ A × V ≤ Λν A × V n+p + n . (3.15)
We let µ and ν 1 , . . . , ν n be the measures defined on B 0 (R) by µ (A) = µ A × V and ν i (A) = ν i A × V for all i = 1, . . . , n. We let ν = n i=1 ν i . By the Lebesgue decomposition of ν with respect to µ, there exist a nonnegative function f in L 1 B 0 (R), d µ and a nonnegative bounded measure σ on B 0 (R) such that there holds ν = f µ + σ and such that σ is singular with respect to µ. We may assume in addition that the function f is identically zero on the support of the measure σ. By (3.15), we get µ x ∈ B 0 (R); f (x) = 0 = 0. For any natural number β, any real number q ≥ 1, and any Borelian set A in B 0 (R), by (3.13)

with ϕ = f q 1 A β , where A β = {x ∈ A ; f (x) ≤ β}, we get A β f qp + d µ ≤ Λ n i=1 A β f qp i d ν i p + np i ≤ Λ n i=1 A β f qp i +1 d µ p + np i ≤ Λ n-n + i=1 A β f qp i +1 d µ p + np i β A β f qp + d µ n + n .
Choosing q large enough so that q > 1/ (p + -p i ) for all i = 1, . . . , n -n + , by Hölder's inequality, it follows that

A β f qp + d µ ≤ β n + n Λν B 0 (R) × V p + q-1 nq n-n + i=1 1 p i - n-n + n A β f qp + d µ 1 nq n-n + i=1 1 p i

+1

.

We then get that either

A β f qp + d µ = 0 or A β f qp + d µ > C β , for some positive constant C β independent of A.
It follows that for any β, the measure A → A β f qp + d µ is a finite linear combination of Dirac masses. Since µ x ∈ B 0 (R); f (x) = 0 = 0, it follows that for any β, the measure A → µ (A β ) is a finite linear combination of Dirac masses. Passing to the limit as β → +∞, we get that there exists an at most countable index set J of distinct points

y j = y j 1 , . . . , y j n-n + in B 0 (R), j ∈ J, such that Supp µ = {y j ; j ∈ J}. It follows that Supp µ ∩ B 0 (R) × V ⊂ j∈J V y j , (3.16)
where V y j = y j 1 , . . . , y j n-n + × V .

(3.17)

We end the proof of Theorem 1.1 by using Palais-Smale properties of the sequence ( u α ) α . For any smooth function φ with compact support in Ω, we get

n i=1 Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i ∂φ ∂x i dx = λ Ω u p + -1 α φdx + o (1) (3.18)
as α → +∞. The functions u p + -1 α keep bounded in L p + /(p + -1) (Ω) and converge, up to a subsequence, almost everywhere to u p + -1 ∞ in Ω as α → +∞. By standard integration theory, it follows that the functions u p + -1 α converge weakly to u p + -1 ∞ in L p + /(p + -1) (Ω). On the other hand, for any i = 1, . . . , n, the functions |∂ u α /∂x i | p i -2 ∂ u α /∂x i keep bounded in L p i /(p i -1) (Ω), and thus converge, up to a subsequence, weakly to a function ψ i in L p i /(p i -1) (Ω) as α → +∞. Passing to the limit into (3.18) as α → +∞, we get

n i=1 Ω ψ i ∂φ ∂x i dx = λ Ω u p + -1 ∞ φdx . (3.19)
By an easy density argument, (3.19) holds true for all functions φ in D 1, - → p (Ω). Now, we let ϕ be a nonnegative, smooth function with support in B 0 (2R) × R n + . Since the sequence ( u α ) α is Palais-Smale for the functional I λ , we get

n i=1 Ω ∂ u α ∂x i p i ϕdx + Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i u α ∂ϕ ∂x i dx = λ Ω u p + α ϕdx + DI λ ( u α ) . ( u α ϕ) ≤ λ Ω u p + α ϕdx + o (1) (3.20)
as α → +∞. For any i = 1, . . . , n -n + , by the compact embeddings in Rákosník [START_REF] Rákosník | Some remarks to anisotropic Sobolev spaces. I[END_REF], we get that the sequence ( u α ) α converges to u ∞ in L p i (Supp ϕ), and thus that

Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i u α ∂ϕ ∂x i dx -→ Ω ψ i u ∞ ∂ϕ ∂x i dx (3.21)
as α → +∞. For any α and any i = n -n + + 1, . . . , n, we get

Ω ∂ u α ∂x i p + -2 ∂ u α ∂x i u α ∂ϕ ∂x i dx ≤ ∂ u α ∂x i p + -1 L p + (Ω) u α L p + (Ω) ∂ϕ ∂x i L ∞ (R n ) . (3.22)
Since the sequence ( u α ) α is bounded in L p + (Ω) and converges to u ∞ almost everywhere in Ω, by Brezis-Lieb [START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF], we get

Ω u p + α ϕdx -→ Ω u p + ∞ ϕdx + B 0 (2R)×V ϕdµ . (3.23)
Since there holds

|∂ u α /∂x i | p i ≥ |∂v α /∂x i | p i -|∂u ∞ /∂x i | p i , where v α = u α -u ∞ for all α and i = 1, . . . , n, we get lim inf α→+∞ Ω ∂ u α ∂x i p i ϕdx ≥ B 0 (2R)×V ϕdν i - Ω ∂u ∞ ∂x i p i ϕdx (3.24)
as α → +∞. By (3.21), (3.22), (3.23), and (3.24), passing to the limit into (3.20) as α → +∞, we get

n i=1 B 0 (2R)×V ϕdν i - n i=1 Ω ∂u ∞ ∂x i p i ϕdx + n-n + i=1 Ω ψ i u ∞ ∂ϕ ∂x i dx ≤ λ Ω u p + ∞ ϕdx + B 0 (2R)×V ϕdµ + C n i=n-n + +1 ∂ϕ ∂x i L ∞ (R n ) (3.25)
for some positive constant C independent of ϕ. Increasing if necessary the constant C, it follows from (3.19) and (3.25) that

n i=1 B 0 (2R)×V ϕdν i - n i=1 Ω ∂u ∞ ∂x i p i ϕdx - n i=1 Ω ψ i ∂u ∞ ∂x i ϕdx ≤ λ B 0 (2R)×V ϕdµ + C n i=n-n + +1 ∂ϕ ∂x i L ∞ (R n ) . (3.26)
We let η be a smooth cutoff function on R n-n + such that η = 1 in B 0 (1), 0 ≤ η ≤ 1 in B 0 (2) \B 0 (1), and η = 0 in R n-n + \B 0 (2). For any point y = y 1 , . . . , y n-n + in B 0 (R) and for any positive real number ε, we let ϕ ε,y be the function defined on R n by

ϕ ε,y (x 1 , . . . , x n ) = η 1 ε (x 1 -y 1 ) , . . . , 1 ε x n-n + -y n-n + .
Plugging ϕ = ϕ ε,y into (3.26), and passing to the limit as ε → 0, we get

n i=1 ν i V y ≤ λµ V y , (3.27)
where V y is as in (3.17). By (3.14) and (3.27), we get that there holds either

µ V y = 0 or λµ V y p + n+p + ≥ Λ -n n+p + (3.28)
for all points y in R n-n + . On the other hand, by (3.11) and by an easy change of variable, for any α, we get

E u α , P - → p y (1) ≤ δ 0 , (3.29)
where the energy functional E is as in (3.1) and P Choosing δ 0 small enough so that δ 0 < Λ

-n p + λ - n+p +
p + , it follows from (3.28) and (3.30) that there holds µ V y = 0 for all points y in B 0 (R). By (3.16), we then get that the measure µ is identically zero on B 0 (R). It follows that |v α | p + ⇀ 0 as α → +∞, where v α = u α -u ∞ , and thus that the sequence ( u α ) α converges to u ∞ in L p + loc (B 0 (R)). This ends the proof of Step 3.2. The next step in the proof of Theorem 1.1 is as follows.

Step 3.3. If the constant δ 0 is small enough, then ∇ u α converges, up to a subsequence, to ∇u ∞ almost everywhere in Ω.

Proof. We let ϕ be a smooth function with compact support in R n . Since the sequence ( u α ) α is Palais-Smale for the functional I λ , there holds DI λ ( u α ) . (( u α -u ∞ ) ϕ) → 0 as α → +∞, and thus

n i=1 Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i ∂ u α ∂x i - ∂u ∞ ∂x i ϕdx + n i=1 Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i ( u α -u ∞ ) ∂ϕ ∂x i dx -λ Ω u p + -1 α ( u α -u ∞ ) ϕdx -→ 0 (3.31)
as α → +∞. By Hölder's inequality and by Step 3.2, we get

Ω u p + -1 α ( u α -u ∞ ) ϕdx ≤ ϕ L ∞ (Ω) u α p + -1 L p + (Ω) u α -u ∞ L p + (Supp ϕ) -→ 0 (3.32)
and

Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i ( u α -u ∞ ) ∂ϕ ∂x i dx ≤ |Supp ϕ| p + -p i p + p i ∂ϕ ∂x i L ∞ (Ω) ∂ u α ∂x i p i -1 L p i (Ω) u α -u ∞ L p + (Supp ϕ) -→ 0 (3.33)
as α → +∞ for all i = 1, . . . , n. By (3.31), (3.32), and (3.33), we get

n i=1 Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i ∂ u α ∂x i - ∂u ∞ ∂x i ϕdx -→ 0 (3.34)
as α → +∞. On the other hand, since the sequence ( u α ) α converges weakly to the function

u ∞ in D 1, - → p (Ω), we get Ω ∂u ∞ ∂x i p i -2 ∂u ∞ ∂x i ∂ u α ∂x i ϕdx -→ Ω ∂u ∞ ∂x i p i ϕdx (3.35)
as α → +∞ for all i = 1, . . . , n. By (3.34) and (3.35), we get

n i=1 Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i - ∂u ∞ ∂x i p i -2 ∂u ∞ ∂x i ∂ u α ∂x i - ∂u ∞ ∂x i ϕdx -→ 0 (3.36)
as α → +∞. Since (3.36) holds true for all smooth functions ϕ with compact support in R n , we then get that for any i = 1, . . . , n and any bounded domain Ω ′ of R n , there holds

Ω ′ ∂ u α ∂x i p i -2 ∂ u α ∂x i - ∂u ∞ ∂x i p i -2 ∂u ∞ ∂x i ∂ u α ∂x i - ∂u ∞ ∂x i dx -→ 0
as α → +∞. In particular, up to a subsequence, there holds

∂ u α ∂x i p i -2 ∂ u α ∂x i - ∂u ∞ ∂x i p i -2 ∂u ∞ ∂x i ∂ u α ∂x i - ∂u ∞ ∂x i -→ 0 a.e. in Ω
as α → +∞. It easily follows that the functions ∂ u α /∂x i converge, up to a subsequence, almost everywhere to ∂u ∞ /∂x i in Ω as α → +∞. This ends the proof of Step 3.3.

The final step in the proof of Theorem 1.1 is as follows.

Step 3.4. The function u ∞ is a nontrivial, nonnegative solution of the problem (1.1).

Proof. We let ϕ be a smooth function with compact support in Ω. Since the sequence ( u α ) α is Palais-Smale for the functional I λ , we get ) keep bounded in L p i /(p i -1) (Ω) (resp. L p + /(p + -1) (Ω)). By standard integration theory, it follows that

n i=1 Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i ∂ϕ ∂x i dx -λ Ω u p + -1 α ϕdx -→ 0 ( 3
Ω u p + -1 α ϕdx -→ Ω u p + -1 ∞ ϕdx (3.38)
and

Ω ∂ u α ∂x i p i -2 ∂ u α ∂x i ∂ϕ ∂x i dx -→ Ω ∂u ∞ ∂x i p i -2 ∂u ∞ ∂x i ∂ϕ ∂x i dx (3.39)
as α → +∞ for all i = 1, . . . , n. By (3.37), (3.38), and (3.39), we get that u ∞ is a solution of problem (1.1). Moreover, u ∞ is nonnegative since the functions u α are nonnegative. We finally claim that u ∞ is not identically zero. Indeed, by (3.11) and by an easy change of variable, for any α, we get E u α , P In particular, u ∞ is not identically zero. This ends the proof of Step 3.4.

Step 3.4 ends the proof of Theorem 1.1

The regularity result

In this section, we prove Theorem 1.2.

Proof of Theorem 1.2. Without loss of generality, we may assume that there exists an index n + such that p n-n + +1 = • • • = p n = p + and p i < p + for all i ≤ n -n + . We let u be a solution of problem (1.1). We begin with proving that u belongs to L q (Ω) for all real numbers q > p + . We let ϕ α = min |u| q-p + p + , α for all positive real numbers α. For any j = 1, . . . , n, multiplying equation (1.1) by uϕ p j α and integrating by parts on Ω, since u = 0 on ∂Ω, we get

n i=1 Ω ∂u ∂x i p i ϕ p j α dx ≤ λ Ω |u| p + ϕ p j α dx . (4.1)
Moreover, for any positive real number β, we get

Ω |u| p + ϕ p j α dx ≤ β (q-p + )p j p + Ω |u| p + dx + W β |u| p + ϕ p j α dx , (4.2)
where

W β = {x ∈ Ω ; |u (x)| > β} . (4.3
) By Hölder's inequality, we get

W β |u| p + ϕ p j α dx ≤ W β |u| p + dx p + -p j p + Ω |u| p + ϕ p + α dx p j p + . (4.4)
Since p + = p * , by the anisotropic Sobolev inequality in Troisi [START_REF] Troisi | Teoremi di inclusione per spazi di Sobolev non isotropi[END_REF], we get

Ω |u| p + ϕ p + α dx ≤ Λ n i=1 Ω ∂u ∂x i p i ϕ p i α dx p + np i (4.5)
for some positive constant Λ independent of α and u. By Young's inequality, it follows that for any ε > 0, there holds

Ω |u| p + ϕ p + α dx ≤ Λ n ε -n + n-n + n-n + i=1 Ω ∂u ∂x i p i ϕ p i α dx p + p i + ε n i=n-n + +1 Ω ∂u ∂x i p + ϕ p + α dx . ( 4
ϕ p i α dx 1 p i ≤ Cβ q-p + p + , (4.10)
where C is independent of α, β, and u. Passing to the limit into (4.10) as α → +∞, we get By the continuity of the embedding of D 1, - → p (Ω) into L p + (Ω), it follows that |u| q p + belongs to L p + (Ω), and thus that u belongs to L q (Ω) for all real numbers q > p + . Now, we prove that u belongs to L ∞ (Ω). For any positive real number t, we define the function ϕ t : R → R by (np + -n-p + )q .

ϕ t (s) =      s + t if s ≤ -
Choosing q large enough so that (p + -1) (nq -n -p + ) (np + -n -p + ) q > 1 , it easily follows that there holds |W t | = 0 for t large, and thus that u belongs to L ∞ (Ω).

Theorem 1 . 2 .

 12 Let n ≥ 3 and -→ p = (p 1 , . . . , p n ), and assume that n i=1 1/p i > 1 and p + = p * . Let Ω be a nonempty, open subset of R n , and λ be a positive real number. Then any solution of problem (1.1) belongs to L ∞ (Ω). Theorem 1.2 is established on arbitrary domains. In case of bounded domains Ω, Theorem 1.2 answers a question posed by Fragalà-Gazzola-Kawohl [24, Section 8.3, Problem 1].

Figure 1 .

 1 Figure 1. Rescaling of a ball (n = 3, p 1 = p 2 = 1.5, p 3 = 6). The first line describes the scale in the rescaling. The second line describes the deformation of the domain.

  is as in(3.2). SinceΩ = R n-n + × V , p n-n + +1 = • • • = p n =p + , and p + = p * , we get τ -→ p µα,xα (Ω) = Ω for all α.

- → p y ( 1 )

 1 is as in(3.10). By(3.23) and since there holds V y ⊂ P -→ p y (1), passing to the limit into (3.29) as α → +∞, it follows that E u ∞ , P

  functional E is as in (3.1) and P -→ p 0 (1) is as in (3.10). By Step 3.2, passing to the limit into (3.40) as α → +∞, we get E u ∞ , P -→ p 0 (1) = δ 0 .

  Choosing ε small enough so that ε < n/ (λΛ), it follows that for some positive constant C independent of α, β, and u. Since the function u belongs to L p + (Ω), increasing if necessary the constant C, it follows from (4.8) and (4.9) that for β large, there holds

		n i=n-n + +1 Ω	∂u ∂x i	p +	ϕ p + α dx ≤	nλβ q-p + n -λΛε Ω	|u| p + dx
											+	λΛ n -λΛε	n-n + i=1	ε	-n + n-n +	Ω	∂u ∂x i	p i	ϕ p i α dx	p + p i	.	(4.8)
	It follows from (4.7) with ε = 1 and (4.8) with ε < n/ (λΛ) that
	n-n + i=1	Ω	∂u ∂x i	p i	ϕ p i α dx	1 p i	≤ C β	q-p + p +		i=1 n-n +		Ω	|u| p + dx	1 p i	(4.9)
	+	n-n + i=1	W β	|u| p + dx	p + -p i p + p i		β	q-p + p +	Ω	|u| p + dx	1 p +	+	n-n + i=1	Ω	∂u ∂x i	p i	ϕ p i α dx	1 p i
									n		∂u	p i
									i=1	Ω	∂x i
																		.6)
	By (4.1)-(4.6), we get
	Ω	∂u ∂x j × p j  ϕ p j α dx ≤ λβ   n-n + i=1 ε -n + (q-p + )p j p + n-n + Ω ∂x i Ω ∂u	|u| p + dx + λ p i ϕ p i α dx p j p i	+	Λ n   ε	p j p + i=n-n + +1 Ω |u| p + dx W β n ∂u ∂x i	p + -p j p + p + ϕ p + α dx	 	p j p +	   . (4.7)

  t , 0 if -t < s < t , s -t if s ≥ t .Multiplying equation (1.1) by ϕ t (u) and integrating by parts on Ω, since u = 0 on ∂Ω, we get Wt |u| p + -2 uϕ t (u) dx , where W t is as in (4.3). For any real number q > p + , by Hölder's inequality, it follows that Since p + = p * , by the anisotropic Sobolev inequality in Troisi[START_REF] Troisi | Teoremi di inclusione per spazi di Sobolev non isotropi[END_REF], and by Young's inequality, we get Wt |ϕ t (u)| p + dx for some positive constant Λ independent of t and u. Since the function u belongs to L q (Ω), it follows from (4.11) and (4.12) that(n+p + )(p + -1)(q-p + ) (np + -n-p + )p + qfor some positive constant C independent of t and u. By Fubini's theorem and Hölder's inequality, we then get

						n n+p +	≤ Λ	n i=1	Wt	∂u ∂x i	p i	dx	p + (n+p + )p i
								≤	p + Λ n + p +	n i=1	1 p i Wt	∂u ∂x i	p i	dx	(4.12)
								1
				|ϕ t (u)| p + dx	p +	≤ C |W t |
				Wt
				n i=1 Wt	∂u ∂x i	p i	dx = λ
	n i=1 Wt	∂u ∂x i	p i	dx ≤ λ |W t |					1 p +	.	(4.11)

(p + -1)(q-p + )

p + q Wt |u| q dx p + -1 q Wt |ϕ t (u)| p + dx +∞ t |W s | ds = +∞ t Wt 1 Ws dxds = Wt |ϕ t (u)| dx ≤ C |W t | (p + -1)(nq-n-p + )

Acknowledgments:

The author was partially supported by the ANR grant ANR-08-BLAN-0335-01. The author is very grateful to Emmanuel Hebey for many helpful remarks and suggestions during the preparation of the manuscript. The author is also very grateful to Alberto Farina for interesting discussions about equations with supercritical growth.