J Ér

Ôme V Étois

Jérôme Vétois
email: vetois@unice.fr

STRONG MAXIMUM PRINCIPLES FOR ANISOTROPIC ELLIPTIC AND PARABOLIC EQUATIONS

We investigate vanishing properties of nonnegative solutions of anisotropic elliptic and parabolic equations. We describe the optimal vanishing sets, and we establish strong maximum principles.

Introduction and results

In dimension n ≥ 2, given -→ p = (p 1 , . . . , p n ) with p i > 1 for i = 1, . . . , n, the anisotropic Laplace operator ∆ - → p is defined by

∆ - → p u = n i=1 ∂ ∂x i ∇ p i x i u , (1.1)
where ∇ p i x i u = |∂u/∂x i | p i -2 ∂u/∂x i . We are concerned with equations of the type ∆ - → p u = f (x, u, ∇u) in Ω (

and

- ∂u ∂t + ∆ - → p u = f (x, t, u, ∇u) in Ω × (0, T ) , (1.3)
where Ω is a domain in R n , T is a positive real number, f is a continuous function, and ∆ - → p is as in (1.1). Anisotropic equations like (1.2) and (1.3) have strong physical background. They emerge, for instance, from the mathematical description of the dynamics of fluids with different conductivities in different directions. We refer to the extensive books by Antontsev-Díaz-Shmarev [START_REF] Antontsev | Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics[END_REF] and Bear [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF] for discussions in this direction. They also appear in biology, see Bendahmane-Karlsen [START_REF] Bendahmane | Renormalized solutions of an anisotropic reaction-diffusion-advection system with L 1 data[END_REF] and 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], as a model describing the spread of an epidemic disease in heterogeneous environments.

In this paper, we investigate strong maximum principles for anisotropic equations of the type (1.2) and (1.3). Given a subset K of Ω, we say that equations (1.2) and (1.3) satisfy a strong maximum principle in K if any nonnegative solution which vanishes at some point in K is in fact identically zero on the whole set K. As is well known (see, for instance, Protter-Weinberger [START_REF] Protter | Maximum principles in differential equations[END_REF]), in case of the standard harmonic and heat equations, namely in case f = 0 and p i = 2 for all i = 1, . . . , n, equations (1.2) and (1.3) satisfy a strong maximum principle in the whole domain Ω.

We show in this paper that in presence of anisotropy, the zeros of solutions may not spread over the whole domain Ω, but they spread along directions where the anisotropic configuration is minimal. We illustrate this fact with a first example. In the anisotropic configuration p 1 = • • • = p n-1 = p -, p n = p + , p -< p + one can check that a nonnegative stationary solution of equations (1.2) and (1.3) with f = 0 on Ω = (0, +∞) n-1 × R is given by

U - → p (x 1 , . . . , x n ) = C |x n | p + /(p + -p -) n-1 i=1 x p -/(p --1) i (p --1)/(p + -p -) , (1.4)
for some constant C = C (n, -→ p ) > 0 under the assumptions that p + > p -(n -2)/(n -1p -) and p -< n -1. The function U - → p vanishes on the set (0, +∞) n-1 × {0} without vanishing elsewhere in the domain. Functions of the form (1.4) were introduced, in a different context, by Giaquinta [START_REF] Giaquinta | Growth conditions and regularity, a counterexample[END_REF] and Marcellini [31]. This example can be generalized by observing that for any C > 0 and ε > 0, the function U - → p satisfies the inequality ∆ - → p u ≤ λu p --1 on Ω = (ε, +∞) n-1 × R for λ > 0 large.

In Theorem 1.1 below, we establish a strong maximum principle for elliptic inequalities of the type ∆ - → p u ≤ f (u) in Ω .

(1.5) In presence of anisotropy, the vanishing sets are of the form

Ω 0 = {x ∈ R n ; [x, ξ 0 ] ⊂ Ω and x i = ξ 0,i ∀i ∈ I + } , (1.6)
for some point ξ 0 = (ξ 0,1 , . . . , ξ 0,n ) in R n , where I + = {i ∈ {1, . . . , n} ; p i > p -} and p -= min (p 1 , . . . , p n ) is the minimum value in the anisotropic configuration. We prove our result under the assumptions that the function f in the right hand sides of (1.5) is continuous, nondecreasing, and such that

f (u) = O u p --1 as u → 0 . (1.7)
We let W 1, - → p loc (Ω) be the Sobolev space defined by

W 1, - → p loc (Ω) = u ∈ L p + loc (Ω) ; ∂u ∂x i ∈ L p i loc (Ω) ∀i = 1, . . . , n ,
where p + = max (p 1 , . . . , p n ) and where, for any real number p ≥ 1, L p loc (Ω) is the space of all measurable functions on Ω which belong to L p (Ω ′ ) for all compact subsets Ω ′ of Ω. We say that a function u in W 1, - → p loc (Ω) ∩ C 0 (Ω) is a (weak) solution of the inequality (1.5) if for any nonnegative smooth function ϕ with compact support in Ω, there holds

- Ω ∂u ∂x i p i -2 ∂u ∂x i ∂ϕ ∂x i dx ≤ Ω f (u) ϕdx .
An historic reference on strong maximum principles for elliptic equations is Hopf [START_REF] Hopf | Elementare Bemerkungen über die lösungen partielle differentialgleichungen zweiter ordnung vom elliptischen typus[END_REF]. We refer to Protter-Weinberger [START_REF] Protter | Maximum principles in differential equations[END_REF] for a reference in book form on this topic. Our first result states as follows.

Theorem 1.1. Let Ω be a nonempty domain in R n and f be a continuous nondecreasing function on R + satisfying (1.7). Let u be a nonnegative solution in W 1, - → p loc (Ω) ∩ C 0 (Ω) of inequality (1.5). If there holds u (ξ 0 ) = 0 for some point ξ 0 in Ω, then the function u is identically zero on the set Ω 0 , where Ω 0 is as in (1.6).

The vanishing sets Ω 0 are the maximal sets on which the strong maximum principle holds true, see (1.4).

Condition (1.7) is optimal among pure nonlinearities of the type f (u) = u p-1 . Indeed, for any real number p in [1, p -), letting i be such that p i = p -, one can check that a nonnegative solution of the equation ∆ - → p u = u p-1 in R n is given by the function p) , where C p,p -= (p -p)/((p

U p,p -(x) = 1 -C p,p -x i p -/(p --
p --1 - p(p --1) 2 ) 1/p -).
Clearly, the functions U p,p -do not satisfy strong maximum principles on sets of the form (1.6).

In Theorem 1.2 below, we establish strong maximum principles for parabolic inequalities of the type

- ∂u ∂t + ∆ - → p u ≤ f (u) in Ω × (0, T ) . (1.8)
We let L -→ p loc (0, T ; W 1, - → p loc (Ω)) be the function space defined by

L - → p loc (0, T ; W 1, - → p loc (Ω)) = u ∈ L p + loc (0, T ; L p + loc (Ω)); ∂u ∂x i ∈ L p i loc (0, T ; L p i loc (Ω)) ∀i = 1, . . . , n ,
where p + = max (p 1 , . . . , p n ) and where, for any real number p ≥ 1, L p loc (0, T ; L p loc (Ω)) is the space of all measurable functions u on Ω × (0, T ) such that t 2 t 1 Ω ′ |u| p dxdt < ∞ for all real numbers 0 < t 1 < t 2 < T and all compact subsets Ω ′ of Ω. We say that a function u in

L - → p loc (0, T ; W 1, - → p loc (Ω)) ∩ C 0 (Ω × (0, T )
) is a (weak) solution of the inequality (1.8) if for any nonnegative smooth function ϕ with compact support in Ω × (0, T ), there holds

T 0 Ω u ∂ϕ ∂t dxdt - T 0 Ω ∂u ∂x i p i -2 ∂u ∂x i ∂ϕ ∂x i dxdt ≤ T 0 Ω f (u) ϕdxdt .
A strong maximum principle for parabolic equations involving the standard Laplace operator was obtained by Nirenberg [START_REF] Nirenberg | A strong maximum principle for parabolic equations[END_REF]. We refer, once again, to the extensive book by Protter-Weinberger [START_REF] Protter | Maximum principles in differential equations[END_REF] on this topic. Our result states as follows.

Theorem 1.2. Let Ω be a nonempty domain in R n , T be a positive real number, and f be a continuous nondecreasing function on R + satisfying (1.7). Let u be a nonnegative solution in

L - → p loc (0, T ; W 1, - → p loc (Ω))∩C 0 (Ω × (0, T )) of inequality (1.8).
Assume that there holds u (ξ 0 , t 0 ) = 0 for some point ξ 0 in Ω and some real number t 0 in (0, T ). Let Ω 0 be as in (1.6). Then we get the following assertions. The vanishing sets in Theorem 1.2 are optimal in the sense that in case p -< 2, we get existence of solutions which extinct in finite time (see Antontsev-Shmarev [START_REF]Parabolic equations with anisotropic nonstandard growth conditions[END_REF][START_REF]Extinction of solutions of parabolic equations with variable anisotropic nonlinearities[END_REF][START_REF]Vanishing solutions of anisotropic parabolic equations with variable nonlinearity[END_REF][START_REF]Localization of solutions of anisotropic parabolic equations[END_REF]), and in case p -> 2, we get existence of solutions which vanish only on a time segment. As an example in case p -> 2, letting i be such that p i = p -, one can consider the function 2) , where

U p -(x, t) = 1 -C p -x i p -/ (1 -(p --2) t) 1/(p --
C p -= (p --2) /(2p p --1 - (p --1) 2 ) 1/p -. As is easily checked, the function U p -is a nonnegative solution of the equation ∂u/∂t = ∆ - → p u in R n × (0, 1/ (p --2)), and we get U p -(x, t) = 0 if and only if x i = 1/C p -.
We refer to Antontsev-Shmarev [START_REF]Parabolic equations with anisotropic nonstandard growth conditions[END_REF][START_REF]Extinction of solutions of parabolic equations with variable anisotropic nonlinearities[END_REF][START_REF]Vanishing solutions of anisotropic parabolic equations with variable nonlinearity[END_REF][START_REF]Localization of solutions of anisotropic parabolic equations[END_REF] for several results on the existence of solutions with finite waiting time or finite extinction time and on the localization of solutions of parabolic equations like (1.3). Other possible references on anisotropic parabolic equations are Antontsev-Chipot [START_REF] Antontsev | Anisotropic equations: uniqueness and existence results[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], and Lieberman [START_REF] Lieberman | Gradient estimates for a new class of degenerate elliptic and parabolic equations[END_REF]. Elliptic equations like (1.2) also received much attention in recent years. Possible references on elliptic equations like (1.2) are Alves-El Hamidi [START_REF] Alves | Existence of solution for a anisotropic equation with critical exponent[END_REF], Antontsev-Shmarev [START_REF] Antontsev | Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions[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],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], 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], Marcellini [START_REF]Regularity and existence of solutions of elliptic equations with p, q-growth conditions[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]Asymptotic stability, convexity, and Lipschitz regularity of domains in the anisotropic regime[END_REF][START_REF]Existence and regularity for critical anisotropic equations with critical directions[END_REF][START_REF]The blow-up of critical anistropic equations with critical directions[END_REF]. We refer to Mercaldo-Rossi-Segura de León-Trombetti [START_REF] Mercaldo | Anisotropic p, q-Laplacian equations when p goes to 1[END_REF] for a description of the asymptotic behavior of solutions of equations like (1.2) as p -→ 1, and we refer to Di Castro-Pérez-Llanos-Urbano [START_REF] Castro | Limits of anisotropic and degenerate elliptic problems[END_REF] and Pérez-Llanos-Rossi [START_REF] Pérez-Llanos | An anisotropic infinity Laplacian obtained as the limit of the anisotropic (p, q)-Laplacian[END_REF] for the case p -→ ∞, where p -= min (p 1 , . . . , p n ) and p + = max (p 1 , . . . , p n ) are the minimum and maximum values in the anisotropic configuration.

In the isotropic configuration where p i = p for all i = 1, . . . , n, the operator (1.1) is comparable, though slightly different, to the p-Laplace operator ∆ p = div |∇u| p-2 ∇u . We refer to Vázquez [START_REF] Vázquez | A strong maximum principle for some quasilinear elliptic equations[END_REF] where the strong maximum principle was established for elliptic equations involving the p-Laplace operator. As for parabolic equations involving the p-Laplace operator, the question of the strong maximum principle was addressed in Nazaret [START_REF] Nazaret | Principe de maximum strict pour un opérateur quasi linéaire[END_REF]. For more material on p-Laplace equations, we refer to the lecture notes by Peral [START_REF] Alonso | Multiplicity of solutions for the p-Laplacian[END_REF].

We also mention the work by Fortini-Mugnai-Pucci [START_REF] Fortini | Maximum principles for anisotropic elliptic inequalities[END_REF] where maximum principles are established for a general class of anisotropic inequalities in divergence form, in particular in the case of variable exponents (see also Zhang [49] concerning this case).

The proofs of Theorems 1.1 and 1.2 rely on the comparison of solutions with a family of anisotropic test functions (see (2.5) and (3.3)). We prove Theorem 1.1 in Section 2, and we prove Theorem 1.2 in Section 3.

Anisotropic elliptic equations

In this section, we prove Theorem 1.1.

Proof of Theorem 1.1. Renumbering, if necessary, the coordinates, we may assume that there exists an index n -such that p 1 = • • • = p n -= p -and p -< p i for all i > n -. We let ξ 0 = (ξ 0,1 , . . . , ξ 0,n ) be a point in Ω such that u (ξ 0 ) = 0. We proceed by contradiction and assume that the function u is not identically zero on Ω 0 , where Ω 0 is as in (1.6). We let P be the set of points x in Ω such that u (x) > 0. Since Ω 0 is arcwise connected and since both the sets P ∩ Ω 0 and Ω 0 \P are nonempty, we get that ∂P ∩ Ω 0 is nonempty. We choose a point

ξ 1 = (ξ 1,1 , . . . , ξ 1,n ) in P ∩ Ω 0 such that inf x∈Ω 0 \P n - i=1 |x i -ξ 1,i | p - p --1 < inf x∈∂Ω n - i=1 |x i -ξ 1,i | p - p --1 , (2.1)
where, by convention, inf ∅ = +∞. Since P is open, it follows from (2.1) that there exist a positive real number r 0 and a point

ζ 0 = (ζ 0,1 , . . . , ζ 0,n ) in Ω 0 \P such that ζ 0 ∈ ∂B p - ξ 1 (r 0 ) and B p - ξ 1 (r 0 )\ {ζ 0 } ⊂ P , where B p - ξ 1 (r 0 ) = x ∈ R n ; n - i=1 |x i -ξ 1,i | p - p --1 < r 0 and x i = ξ 0,i ∀i > n - (2.2)
and

∂B p - ξ 1 (r 0 ) = x ∈ R n ; n - i=1 |x i -ξ 1,i | p - p --1 = r 0 and x i = ξ 0,i ∀i > n -. (2.3)
For any positive real numbers δ and ε, we let A -→ p ξ 1 (r 0 , δ, ε) be the annular set defined by

A - → p ξ 1 (r 0 , δ, ε) = x ∈ R n ; r 0 -ε < n i=1 δ p --p i p i -1 |x i -δζ 0,i -(1 -δ) ξ 1,i | p i p i -1 < r 0 . (2.4)
Since B p - ξ 1 (r 0 )\ {ζ 0 } ⊂ P , we get that for δ small and any ε, A -→ p ξ 1 (r 0 , δ, ε) is included in Ω. Moreover, we get that for ε fixed and δ small, the point ζ 0 belongs to A -→ p ξ 1 (r 0 , δ, ε). For any positive real numbers λ and δ, we define our test function v λ,δ on R n by

v λ,δ (x) = λδ e λ r 0 -n i=1 (λ 2 δ) p --p i p i -1 |x i -δζ 0,i -(1-δ)ξ 1,i | p i p i -1 -1 .
(2.5)

Letting ∆ - → p be as in (1.1), we find

∆ - → p v λ,δ (x) = λ 2 δ p --1 n i=1 p i p i -1 p i -1 e (p i -1)λ r 0 -n j=1 (λ 2 δ) p --p j p j -1 |x j -δζ 0,j -(1-δ)ξ 1,j | p j p j -1 × p i λ 2p --p i -1 p i -1 δ p --p i p i -1 |x i -δζ 0,i -(1 -δ) ξ 1,i | p i p i -1 -1 . (2.6)
For any point x in A -→ p ξ 1 (r 0 , λ 2 δ, ε), by (1.7), (2.5), and (2.6), we get

-∆ - → p v λ,δ (x) + f (v λ,δ (x)) ≤ (λδ) p --1 n i=1 p i p i -1 p i -1 λ p --1 e (p i -1)λε - p p - -λ p -(r 0 -ε) (p --1) p --1 + C e λε -1 p --1 (2.7)
when λδ and λε are small, for some positive constant C independent of λ, δ, ε, and x. Choosing λ large enough so that

λ > (p --1) p --1 p p - -r 0 n i=1 p i p i -1 p i -1
, and then, choosing δ and ε small, it follows from (2.7) that v λ,δ is a C 1 -solution of the inequality

-∆ - → p v λ,δ + f (v λ,δ ) < 0 in A - → p ξ 1 r 0 , λ 2 δ, ε , (2.8)
where

A - → p ξ 1 (r 0 , λ 2 δ, ε) is as in (2.4). We let ∂ 1 A - → p ξ 1 (r 0 , λ 2 δ, ε) and ∂ 2 A - → p
ξ 1 (r 0 , λ 2 δ, ε) stand for the respective interior and exterior boundaries of the annular set A -→ p ξ 1 (r 0 , λ 2 δ, ε). Since the function u is positive on B p - ξ 1 (r 0 ), by continuity, we get the existence of a positive constant

C ε such that u > C ε on B p - ξ 1 (r 0 -ε), where B p - ξ 1 (r 0 -ε) is as in (2.2). Still by continuity of u, it follows that u ≥ C ε on ∂ 1 A - → p ξ 1 (r 0 , λ 2 δ, ε) for δ small. Since v λ,δ = λδ e λε -1 on ∂ 1 A - → p ξ 1 (r 0 , λ 2 δ, ε) and v λ,δ = 0 on ∂ 2 A - → p ξ 1 (r 0 , λ 2 δ, ε), we then get v λ,δ ≤ u on ∂A - → p ξ 1 (
r 0 , λ 2 δ, ε) for δ small. In particular, there holds (v λ,δu) + = 0 on ∂A -→ p ξ 1 (r 0 , λ 2 δ, ε), where (v λ,δu) + = max (v λ,δu, 0). Testing (1.5) and (2.8) against (v λ,δu) + and integrating by parts on A -→ p ξ 1 (r 0 , λ 2 δ, ε), we then get

n i=1 W λ,δ,ε ∂v λ,δ ∂x i p i -2 ∂v λ,δ ∂x i - ∂u ∂x i p i -2 ∂u ∂x i ∂v λ,δ ∂x i - ∂u ∂x i dx + W λ,δ,ε (f (v λ,δ ) -f (u)) (v λ,δ -u) dx ≤ 0 , (2.9)
where

W λ,δ,ε = x ∈ A - → p ξ 1 r 0 , λ 2 δ, ε ; v λ,δ (x) > u (x) .
Since the function f is nondecreasing, it follows from (2.9) that

n i=1 W λ,δ,ε ∂v λ,δ ∂x i p i -2 ∂v λ,δ ∂x i - ∂u ∂x i p i -2 ∂u ∂x i ∂v λ,δ ∂x i - ∂u ∂x i dx = 0 ,
and thus that ∇u = ∇v λ,δ almost everywhere in W λ,δ,ε . Since W λ,δ,ε is open, we then get that the function v λ,δu is constant in W λ,δ,ε . By continuity of u and v λ,δ , it follows that

|W λ,δ,ε | = 0, i.e. v λ,δ ≤ u in A - → p ξ 1 (r 0 , λ 2 δ, ε). In particular, we get u (ζ 0 ) ≥ v λ,δ (ζ 0 ) > 0.
There is a contradiction. This ends the proof of Theorem 1.1.

Anisotropic parabolic equations

This section is devoted to the proof of Theorem 1.2. Renumbering, if necessary, the coordinates, we may assume in what follows that there exists an index n -such that p 1 = • • • = p n -= p -and p -< p i for all i > n -. For any positive real numbers µ, r, and any point (ξ, t) in R n × R + , we define the sets

B p - (ξ,t) (µ, r) and ∂B p - (ξ,t) (µ, r) by B p - (ξ,t) (µ, r) = (x, s) ∈ R n × R + ; n - i=1 |x i -ξ i | p - p --1 + µ |s -t| p - p --1 < r and x i = ξ i ∀i > n - (3.1) and ∂B p - (ξ,t) (µ, r) = (x, s) ∈ R n × R + ; n - i=1 |x i -ξ i | p - p --1 + µ |s -t| p - p --1 = r and x i = ξ i ∀i > n -. (3.2)
As a preliminary step in the proof of Theorem 1.2, we prove the following lemma. Lemma 3.1. Let Ω, T , f , and u be as in Theorem 1.2. Let µ be a positive real number. Assume that there exist a positive real number r 0 and two points (ξ 0 , t 0 ) and (ξ 1 , t 1 ) in Ω×(0, T ) such that u (ξ 0 , t 0 ) = 0, (ξ 0 , t 0 ) ∈ ∂B p - (ξ 1 ,t 1 ) (µ, r 0 ), B p - (ξ 1 ,t 1 ) (µ, r 0 ) ⊂ Ω 0 × (0, T ), and u (x, t) > 0 for all points (x, t) in B p - (ξ 1 ,t 1 ) (µ, r 0 ) \ {(ξ 0 , t 0 )}, where Ω 0 is as in (1.6), B p - (ξ 1 ,t 1 ) (µ, r 0 ) is as in (3.1), and ∂B p - (ξ 1 ,t 1 ) (µ, r 0 ) is as in (3.2). Then we get the following assertions.

(i) If p -≤ 2, then ξ 0 = ξ 1 . (ii) If p -= 2 and µ > 1 4r 0 n i=1 p i p i -1 p i -1 2 , then t 0 = t 1 -r 0 /µ. (iii) If p -> 2, then t 0 ≤ t 1 .
Proof of Lemma 3.1. We proceed by contradiction and assume that ξ 0 = ξ 1 if p -< 2, either ξ 0 = ξ 1 or t 0 > t 1 if p -= 2, and t 0 > t 1 if p -> 2. Moreover, decreasing, if necessary, the real number r 0 , we may assume that u (x, t) > 0 for all points (x, t) on ∂B

p - (ξ 1 ,t 1 ) (µ, r 0 ) \ {(ξ 0 , t 0 )}, where ∂B p - (ξ 1 ,t 1 ) (µ, r 0 ) is as in (3.
2). For any positive real numbers λ, µ, and δ, we define our

test function v λ,µ,δ on R n × R + by v λ,µ,δ (x) = λδ e λ r 0 -n i=1 (λ 2 δ) p --p i p i -1 |x i -δξ 0,i -(1-δ)ξ 1,i | p i p i -1 -µ|t-δt 0 -(1-δ)t 1 | p - p --1 -1 , (3.3)
where ξ 0 = (ξ 0,1 , . . . , ξ 0,n ) and ξ 1 = (ξ 1,1 , . . . , ξ 1,n ). We find

∂v λ,µ,δ ∂t (x, t) = p - p --1 µλ 2 δe λ r 0 -n i=1 (λ 2 δ) p --p i p i -1 |x i -δξ 0,i -(1-δ)ξ 1,i | p i p i -1 -µ|t-δt 0 -(1-δ)t 1 | p - p --1 × |δt 0 + (1 -δ) t 1 -t| 2-p - p --1 (δt 0 + (1 -δ) t 1 -t) . (3.4)
Moreover, letting ∆ - → p be as in (1.1), we find

∆ - → p v λ,µ,δ (x, t) = λ 2 δ p --1 n i=1 p i p i -1 p i -1 × e (p i -1)λ r 0 -n j=1 (λ 2 δ) p --p j p j -1 |x j -δξ 0,j -(1-δ)ξ 1,j | p j p j -1 -µ|t-δt 0 -(1-δ)t 1 | p - p --1 × p i λ 2p --p i -1 p i -1 δ p --p i p i -1 |x i -δξ 0,i -(1 -δ) ξ 1,i | p i p i -1 -1 . (3.5)
As is easily seen, for δ small, for any i = 1, . . . , n and any point (x, t) in R n × R + , there holds

|x i -δξ 0,i -(1 -δ) ξ 1,i | p i p i -1 -|ξ 0,i -ξ 1,i | p i p i -1 ≤ C |ξ 0,i -ξ 1,i | 1 p i -1 |x i -ξ 0,i | + |x i -ξ 0,i | p i p i -1 + δ |ξ 0,i -ξ 1,i | p i p i -1 , (3.6) |t -δt 0 -(1 -δ) t 1 | p - p --1 -|t 0 -t 1 | p - p --1 ≤ C |t 0 -t 1 | 1 p --1 |t -t 0 | + |t -t 0 | p - p --1 + δ |t 0 -t 1 | p - p --1 , (3.7)
and

|t -δt 0 -(1 -δ) t 1 | 2-p - p --1 (t -δt 0 -(1 -δ) t 1 ) -|t 0 -t 1 | 2-p - p --1 (t 0 -t 1 ) ≤      C |t 0 -t 1 | 2-p - p --1 |t -t 0 | + |t -t 0 | 1 p --1 + δ 0 -t 1 | 1 p --1 if p -≤ 2 C |t -t 0 | 1 p --1 + δ 1 p --1 |t 0 -t 1 | 1 p --1 if p -> 2 (3.8)
for some positive constant C independent of δ, x, and t. For any positive real numbers µ, δ, and ε, we define the ellipsoidal ball

B - → p (ξ 0 ,t 0 ) (µ, δ, ε) by B - → p (ξ 0 ,t 0 ) (µ, δ, ε) = (x, s) ∈ R n × R + ; n i=1 δ p --p i p i -1 |x i -ξ 0,i | p i p i -1 + µ |t -t 0 | p - p --1 < ε . (3.9)
Clearly, for µ large and for δ and ε small, B -→ p (ξ 0 ,t 0 ) (µ, δ, ε) is included in Ω × (0, T ). For any positive real numbers λ, µ, δ, ε, and any point (x, t) in B -→ p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε), by (1.7), (3.3)-(3.8), and since (ξ

0 , t 0 ) ∈ ∂B p - (ξ 1 ,t 1 ) (µ, r 0 ), we get ∂v λ,µ,δ ∂t (x, t) ≤ p - p --1 µλ 2 δ (3.10) ×                                  e Cλ(µ+1) ε p --1 p -+δ |t 1 -t 0 | 2-p - p --1 (t 1 -t 0 ) + C ε p --1 p - + δ if p -≤ 2 and t 0 ≤ t 1 e -Cλ(µ+1) ε p --1 p -+δ |t 1 -t 0 | 2-p - p --1 (t 1 -t 0 ) + Ce Cλ(µ+1) ε p --1 p -+δ ε p --1 p - + δ if p -≤ 2 and t 0 > t 1 e -Cλ(µ+1) ε p --1 p -+δ |t 1 -t 0 | 2-p - p --1 (t 1 -t 0 ) + Ce Cλ(µ+1) ε p --1 p -+δ ε 1 p -+ δ 1 p --1 if p -> 2 and t 0 > t 1 and -∆ - → p v λ,µ,δ (x, t)+f (v λ,µ,δ (x, t)) ≤ (λδ) p --1 n i=1 p i p i -1 p i -1 λ p --1 e (p i -1)Cλ(µ+1) ε p --1 p -+δ - p p - -λ p - (p --1) p --1 e -(p + -1)Cλ(µ+1) ε p --1 p -+δ r 0 -µ |t 1 -t 0 | p - p --1 + Ce Cλ(µ+1) ε p --1 p -+δ ε p --1 p - + δ + C e Cλ(µ+1) ε p --1 p -+δ -1 p --1 (3.11)
when δ, ε, and λ (µ + 1) ε (p --1)/p -+ δ are small, for some positive constant C independent of λ, µ, δ, ε, x, and t. In case p -≤ 2 and ξ 0 = ξ 1 , since (ξ 0 , t 0 ) ∈ ∂B p - (ξ 1 ,t 1 ) (µ, r 0 ), we get µ |t 1t 0 | p -/(p --1) < r 0 . We choose λ large enough so that

             λ > (p --1) p --1 p p - - r 0 -µ |t 1 -t 0 | p - p --1 n i=1 p i p i -1 p i -1 if p -< 2 and ξ 0 = ξ 1 λ > 1 4 r 0 -µ (t 1 -t 0 ) 2 2µ (t 1 -t 0 ) + n i=1 p i p i -1 p i -1 if p -= 2 and ξ 0 = ξ 1 (3.12)
It follows from (3.10), (3.11), and (3.12) that in case p -≤ 2 and ξ 0 = ξ 1 , for δ and ε small, the function v λ,µ,δ is a C 1 -solution of the inequality

∂v λ,µ,δ ∂t -∆ - → p v λ,µ,δ + f (v λ,µ,δ ) < 0 in B - → p (ξ 0 ,t 0 ) µ, λ 2 δ, ε , (3.13)
where B -→ p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε) is as in (3.9). In case p -= 2 and t 0 = t 1 + r 0 /µ, we assume that µ > 1 4r 0 n i=1 p i p i -1 p i -1 2 , we let λ be an arbitrary positive real number, and we also find (3.13) for δ and ε small. In case p -> 2 and t 0 > t 1 , without assumption on λ and µ, we still get (3.13) for δ and ε small. Now, we claim that there exists a positive constant

C ε such that u ≥ C ε on B - → p (ξ 1 ,t 1 ) (µ, λ 2 δ, r 0 ) ∩ ∂B - → p
(ξ 0 ,t 0 ) (µ, λ 2 δ, ε) for δ small, where B -→ p (ξ 1 ,t 1 ) (µ, λ 2 δ, r 0 ) and B -→ p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε) are as in (3.9). In order to prove this claim, we proceed by contradiction and assume that there exist a sequence of positive real numbers (δ α ) α and a sequence of points (ξ α , t α ) α such that δ α → 0, u (ξ α , t α ) → 0 as α → +∞, and (ξ α , t α ) ∈ B -→ p (ξ 1 ,t 1 ) (µ, λ 2 δ α , r 0 ) ∩ ∂B -→ p (ξ 0 ,t 0 ) (µ, λ 2 δ α , ε) for all α. Up to a subsequence, we get that (ξ α , t α ) converges to a point (ξ ∞ , t ∞ ) in B p - (ξ 1 ,t 1 ) (µ, r 0 )∩∂B p - (ξ 0 ,t 0 ) (µ, ε), where B p - (ξ 1 ,t 1 ) (µ, r 0 ) and ∂B p - (ξ 0 ,t 0 ) (µ, ε) are as in (3.1) and (3.2). By continuity of the function u, we get u (ξ ∞ , t ∞ ) = 0, and thus (ξ ∞ , t ∞ ) = (ξ 0 , t 0 ). Since (ξ α , t α ) ∈ B -→ p (ξ 1 ,t 1 ) (µ, λ 2 δ α , r 0 ) for all α and since ξ 1,i = ξ 0,i for all i > n -, it follows that (3.14) as α → +∞. On the other hand, since (ξ α , t α ) ∈ ∂B -→ p (ξ 0 ,t 0 ) (µ, λ 2 δ α , ε) for all α, we get

n i=n -+1 λ 2 δ α p --p i p i -1 |ξ α,i -ξ 0,i | p i p i -1 < r 0 -µ |t α -t 1 | p - p --1 - n - i=1 |ξ α,i -ξ 1,i | p - p --1 = o (1)
n i=n -+1 λ 2 δ α p --p i p i -1 |ξ α,i -ξ 0,i | p i p i -1 = ε -µ |t α -t 0 | p - p --1 - n - i=1 |ξ α,i -ξ 0,i | p - p --1 = ε + o (1) (3.15)
as α → +∞. There is a contradiction between (3.14) and (3.15). This ends the proof of our claim, namely that there exists a positive constant

C ε such that u ≥ C ε on B - → p (ξ 1 ,t 1 ) (µ, λ 2 δ, r 0 ) ∩ ∂B - → p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε) for δ small. Since v λ,µ,δ ≤ λδ e λr 0 -1 in B - → p (ξ 1 ,t 1 ) (µ, λ 2 δ, r 0 ) and v λ,µ,δ ≤ 0 in (R n × R + ) \B - → p (ξ 1 ,t 1 ) (µ, λ 2 δ, r 0 ), we then get v λ,µ,δ ≤ u on ∂B - → p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε) for δ small.
In particular, there holds (v λ,µ,δu) + = 0 on ∂B -→ p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε), where (v λ,µ,δu) + = max (v λ,µ,δu, 0). Testing (1.8) and (3.13) against (v λ,µ,δu) + on B -→ p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε) (up to an approximation in terms of Steklov averages, see, for instance, DiBenedetto [START_REF] Dibenedetto | Degenerate parabolic equations[END_REF]), we get

1 2 W λ,µ,δ,ε,t |v λ,µ,δ -u| 2 dx + n i=1 t 0 W λ,µ,δ,ε,s ∂v λ,µ,δ ∂x i p i -2 ∂v λ,µ,δ ∂x i - ∂u ∂x i p i -2 ∂u ∂x i ∂v λ,µ,δ ∂x i - ∂u ∂x i dxds + t 0 W λ,µ,δ,ε,s (f (v λ,µ,δ ) -f (u)) (v λ,µ,δ -u) dxds ≤ 0 (3.16)
for all real numbers t in (0, T ), where

W λ,µ,δ,ε,t = x ∈ R n ; (x, t) ∈ B - → p (ξ 0 ,t 0 ) µ, λ 2 δ, ε and v λ,µ,δ (x, t) > u (x, t) .
Since the function f is nondecreasing, it follows from (3.16) that for any real number t in (0, T ), there holds

W λ,µ,δ,ε,t |v λ,µ,δ -u| 2 dx = 0 . We then get |W λ,µ,δ,ε,t | = 0, i.e. v λ,µ,δ ≤ u in B - → p (ξ 0 ,t 0 ) (µ, λ 2 δ, ε).
In particular, we get u (ξ 0 , t 0 ) ≥ v λ,µ,δ (ξ 0 , t 0 ) > 0. There is a contradiction. This ends the proof of Lemma 3.1. Now, we can prove Theorem 1.2 by using Lemma 3.1.

Proof of Theorem 1.2. To begin with, we assume that p -≤ 2 and prove that the function u is identically zero on the set Ω 0 × {t 0 }, where Ω 0 is as in (1.6). We let P be the set of points (x, t) in Ω × (0, T ) such that u (x, t) > 0. We proceed by contradiction and assume that P ∩ (Ω 0 × {t 0 }) is not empty. In a similar way as in the proof of Theorem 1.1, we can choose a positive real number r 0 and two points ζ 0 = (ζ 0,1 , . . . , ζ 0,n ) and ξ 1 = (ξ 1,1 , . . . , ξ 1,n )

in Ω 0 such that u (ζ 0 , t 0 ) = 0, ζ 0 ∈ ∂B (3.17)

Since u (ζ 0 , t 0 ) = 0, we get h (δ) → 0 as δ → 0. In particular, we get ∂B p - (δξ 1 +(1-δ)ζ 0 ,t 0 ) (1, h (δ)) ⊂ Ω 0 × (0, T ) and B p - (δξ 1 +(1-δ)ζ 0 ,t 0 ) (1, h (δ)) ⊂ P for δ small, where B p - (δξ 1 +(1-δ)ζ 0 ,t 0 ) (1, h (δ)) and ∂B p - (δξ 1 +(1-δ)ζ 0 ,t 0 ) (1, h (δ)) are as in (3.1) and (3.2). Since P is open, it follows that for δ small, the infimum in (3.17) is achieved, i.e. there exists a point (ζ δ , t δ ) on ∂B p - (δξ 1 +(1-δ)ζ 0 ,t 0 ) (1, h (δ)) such that u (ζ δ , t δ ) = 0. By Lemma 3.1, we get ζ δ = δξ 1 + (1δ) ζ 0 , and thus h (δ) = |t δt 0 | p -/(p --1) for δ small. It follows from (3.17) that for δ 1 and δ 2 small, there holds

h (δ 1 ) ≤ n - i=1 |ξ 1,i -ζ 0,i | p - p --1 |δ 2 -δ 1 | p - p --1 + h (δ 2 ) .
In particular, for δ small, the function h is differentiable and h ′ = 0 on [0, δ]. It follows that the function h is constant on [0, δ]. Since h (0) = 0, we then get h = 0 on [0, δ], i.e. u = 0 on [δξ 1 + (1δ) ζ 0 , ζ 0 ] × {t 0 }. There is a contradiction. This ends the proof of the first part of Theorem 1.2. Now, we assume that p -≥ 2, and we prove that the function u is identically zero on the set {ξ 0 } × (0, t 0 ]. We proceed by contradiction and assume that there exists a real number t 1 in (0, t 0 ) such that u (ξ 0 , t 1 ) > 0. Since P is open, we get B p - (ξ 0 ,t 1 ) (µ, ε) ⊂ P for µ large and ε small, where B p - (ξ 0 ,t 1 ) (µ, ε) is as in (3.1). We may assume that the real number µ is large enough so that µ > 1 2ε n i=1 p i p i -1 p i -1 . Increasing, if necessary, the real number t 1 , since P is open, we may assume that t 1 = sup t ∈ (0, t 0 ) ; B p - (ξ 0 ,t 1 ) (µ, ε) ⊂ P .

It follows that there exists a point (ξ 2 , t 2 ) on ∂B p - (ξ 0 ,t 1 ) (µ, ε) such that t 2 > t 1 and u (ξ 2 , t 2 ) = 0. We get a contradiction with Lemma 3.1. This ends the proof of Theorem 1.2.

Date:

  April 15, 2011. Published in Advanced Nonlinear Studies 12 (2012), no. 1, 101-114.

  (i) If p -< 2, then the function u is identically zero on the set Ω 0 × {t 0 }. (ii) If p -= 2, then the function u is identically zero on the set Ω 0 × (0, t 0 ]. (iii) If p -> 2, then the function u is identically zero on the set {ξ 0 } × (0, t 0 ].

p - ξ 1

 1 (r 0 ), and B p - ξ 1 (r 0 ) × {t 0 } ⊂ P , where B p - ξ 1 (r 0 ) and ∂B p - ξ 1 (r 0 ) are as in (2.2) and (2.3). We let h : [0, 1] → R + be defined byh (δ) = inf (x,t)∈(Ω 0 ×(0,T ))\P n - i=1 |x iδξ 1,i -(1δ) ζ 0,i | pp --1 + |tt 0 | pp --1.

Acknowledgments:

The author was partially supported by the ANR grant ANR-08-BLAN-0335-01.


Jérôme Vétois
email: vetois@unice.fr

THE BLOW-UP OF CRITICAL ANISTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

Introduction

Anisotropic operators modelize directionally dependent phenomena. In this paper, we consider problems posed on domains in the Euclidean space R n in dimension n ≥ 2, and we provide ourselves with an anisotropic configuration -→ p = (p 1 , . . . , p n ) with p i > 1 for all i = 1, . . . , n. We define the anisotropic Laplace operator ∆ - → p by

∆ - → p u = n i=1 ∂ ∂x i ∇ p i x i u , (1.1)
where ∇ p i x i u = |∂u/∂x i | p i -2 ∂u/∂x i for all i = 1, . . . , n. Nonlinear equations of the type ∆ - → p u = f (•, u) appear in several places in the literature. They appear, for instance, in biology, see Bendahmane-Karlsen [START_REF] Bendahmane | Renormalized solutions of an anisotropic reaction-diffusion-advection system with L 1 data[END_REF] and 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], as a model describing the spread of an epidemic disease in heterogeneous environments. They also emerge, see Antontsev-Díaz-Shmarev [START_REF] Antontsev | Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics[END_REF] and Bear [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF], from the mathematical description of the dynamics of fluids with different conductivities in different directions.

We consider anisotropic problems of critical growth of the type

-∆ - → p u = λ |u| p * -2 u + f (•, u) in Ω , u ∈ D 1, - → p (Ω) , (1.2)
where ∆ - → p is as in (1.1), Ω is a domain of R n , D 1, - → p (Ω) is 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) , p * is the critical Sobolev exponent (see (1.4) below), λ is a positive real number, and f is a Caratheodory function in Ω × R satisfying the growth condition

|f (•, u)| ≤ C |u| q-1 + 1 a.e. in Ω (1.3)
for some real number q in (1, p * ) and for some positive constant C independent of u.

We are concerned with the doubly critical situation p + = p * , where p + = max (p 1 , . . . , p n ) is the maximum value of the anisotropic configuration and p * is as above 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 (1.2).

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
.

(1.4)

Possible references on anisotropic Sobolev spaces are Besov [START_REF] Besov | Embeddings of an anisotropic Sobolev space for a domain with a flexible horn condition[END_REF], Haškovec-Schmeiser [START_REF] Haškovec | A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems[END_REF], Kruzhkov-Kolodīȋ [START_REF] Kruzhkov | On the theory of anisotropic Sobolev spaces[END_REF], Kruzhkov-Korolev [START_REF] Kruzhkov | On embedding theory for anisotropic function spaces[END_REF], Lu [START_REF] Lu | On imbedding theorems for spaces of functions with partial derivatives of various degrees of summability[END_REF], Nikol ′ skiȋ [START_REF] Nikol | Imbedding, continuation and approximation theorems for differentiable functions of several variables[END_REF], Rákosník [START_REF] Rákosník | Some remarks to anisotropic Sobolev spaces. I[END_REF][START_REF]Some remarks to anisotropic Sobolev spaces. II[END_REF], and Troisi [START_REF] Troisi | Teoremi di inclusione per spazi di Sobolev non isotropi[END_REF].

We aim in describing the asymptotic behaviors in energy space of Palais-Smale sequences associated with problem (1.2). Before stating our main result, let us fix some notations. For any µ > 0 and any point a = (a 1 , . . . , a n ) in R n , we define the affine transformation τ (x 1a 1 ) , . . . , µ pn-p * pn (x na n ) .

(1.5)

As is easily checked, (1.5) provides a general rescaling invariance rule associated with problem (1.2) with f ≡ 0. In particular, u solves -∆ - → p u = |u| p * -2 u in Ω if and only if the function

v = µ -1 u • τ - → p µ,a -1 solves -∆ - → p v = |v| p * -2 v in τ - → p µ,a (Ω), 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 .
Given (µ α ) α a sequence of positive real numbers converging to 0, (x α ) α a converging sequence in R n , λ a positive real number, U a nonempty, open subset of R n , and u a nontrivial solution in D 1, - → p (U ) of the problem

-∆ - → p u = λ |u| p * -2 u in U , u ∈ D 1, - → p (U ) , (1.6)
where ∆ - → p is as in (1.1), we call -→ p -bubble of centers (x α ) α , weights (µ α ) α , multiplier λ, domain U , and profile u, the sequence (B α ) α defined by

B α = 1 µ α u • τ - → p µα,xα
for all α, where τ -→ p µα,xα is as in (1.5). One can find existence and regularity results for problem (1.6) in Vétois [START_REF]Existence and regularity for critical anisotropic equations with critical directions[END_REF]. In the following, we implicitly extend profiles of bubbles by 0 outside of their domains so as to regard them as functions in D 1, - → p (R n ). With the above notations, we

define the energy E (B α ) of a -→ p -bubble (B α ) α by E (B α ) = n i=1 1 p i R n ∂u ∂x i p i dx - λ p * R n |u| p * dx = n i=1 p * -p i p * p i R n ∂u ∂x i p i dx , (1.7)
the second equality in (1.7) being obtained by testing (1.6) with u and integrating by parts.

We approximate problem (1.2) with the problems

-∆ - → p u = λ α |u| rα-2 u + f (•, u) in Ω , u ∈ D 1, - → p (Ω) , (1.8)
where ∆ - → p is as in (1.1), (λ α ) α is a sequence of positive real numbers converging to λ, and (r α ) α is a sequence of real numbers in (1, p * ] converging to p * . Associated with problems (1.2) and (1.8), we define the functionals I α and I ∞ in D 1, - → p (Ω) by

I α (u) = n i=1 1 p i Ω ∂u ∂x i p i dx - Ω F (x, u) dx - λ α r α Ω |u| rα dx (1.9)
and

I ∞ (u) = n i=1 1 p i Ω ∂u ∂x i p i dx - Ω F (x, u) dx - λ p * Ω |u| p * dx , (1.10)
where

F (x, u) = u 0 f (x, s) ds. We say that a sequence (u α ) α in D 1, - → p (Ω) is Palais-Smale for the functionals (I α ) α if there hold |I α (u α )| ≤ C for some positive constant C independent of α and DI α (u α ) → 0 in D 1, - → p (Ω) ′ as α → +∞. We say that (u α ) α is a bounded Palais-Smale sequence for the functionals (I α ) α if (u α ) α is moreover bounded in D 1, - → p (Ω).
Theorem 1.1 below is referred to as a bubble tree decomposition. With this result, we provide a complete description of the asymptotic behaviors in D 1, - → p (Ω) of bounded Palais-Smale sequences for the functionals (I α ) α defined in (1.9). Bubble tree decompositions were obtained by Struwe [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF] for the equation -∆u = u 2 * -1 , where ∆ is the classical Laplace operator and 2 * = 2n/ (n -2) is the critical Sobolev exponent. Related references to Struwe [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF] are Brézis-Coron [START_REF] Brézis | Convergence of solutions of H-systems or how to blow bubbles[END_REF], 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], Sacks-Uhlenbeck [START_REF] Sacks | The existence of minimal immersions of 2-spheres[END_REF], Schoen [START_REF] Schoen | Variational theory for the total scalar curvature functional for Riemannian metrics and related topics[END_REF], and Wente [START_REF] Wente | Large solutions to the volume constrained Plateau problem[END_REF]. We also refer to El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF] for an extension of Struwe's result to the anisotropic Laplace operator in case p + < p * . Here, we treat the more involved case of critical directions p + = p * . The definition of asymptotically -→ p -stable domains is postponed to Section 3. Roughly speaking, asymptotically stable domains are domains which, in the limit of the rescalings, still satisfy the segment property. This notion, see El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF] and Vétois [START_REF]Asymptotic stability, convexity, and Lipschitz regularity of domains in the anisotropic regime[END_REF], turns out to be fundamentally associated with the question of proving bubble tree decompositions for anisotropic critical equations. We state our result as follows.

Theorem 1.1. 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 an asymptotically -→ p -stable, bounded domain of R n , and f be a Caratheodory function in Ω ×R satisfying the growth condition (1.3). Let (λ α ) α be a sequence of positive real numbers converging to λ, and (r α ) α be a sequence of real numbers in (1, p + ] converging to p + . For any bounded Palais-Smale sequence (u α ) α for the functionals (I α ) α defined in (1.9), there exist a solution u ∞ of problem (1.2), a natural number k, and for any j = 1, . . . , k, a -→ p -bubble (B j α ) α of weights (µ j α ) α and multiplier λ j satisfying (µ j α ) p + -rα → λ j /λ as α → +∞, such that, up to a subsequence, there holds

u α = u ∞ + k j=1 B j α + R α (1.11)
for all α, where R α → 0 in D 1, - → p (R n ) as α → +∞. Moreover, there holds

I α (u α ) = I ∞ (u ∞ ) + k j=1 E B j α + o (1) (1.12)
as α → +∞, where E, I α , and I ∞ are as in (1.7), (1.9), and (1.10). If in addition the functions u α are nonnegative, then u ∞ and (B 1 α ) α , . . . , B k α α are also nonnegative. We give examples in Section 2, see Corollary 2.5, of Palais-Smale sequences which blow up, namely which develop at least one -→ p -bubble in their decompositions.

The analogue of Theorem 1.1 for the case p + < p * was established in El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF]. The essential difference in our doubly critical setting p + = p * lies in the effect of the transformation (1.5). Indeed, contrary to the case p + < p * where all directions are subcritical, the domain is only rescaled with respect to some directions, the noncritical ones, in case p + = p * . Therefore, we observe a concentration phenomenon on affine subspaces of R n spanned by critical directions. For instance, Figure 1 below illustrates the rescaling of a three-dimensional ball, the first two directions being noncritical and the third direction being critical. Together with the rescaling of the domain, the equation in (1.2) is rescaled. At the limit, we find the purely invariant problem (1.6). This problem is studied in Vétois [START_REF]Existence and regularity for critical anisotropic equations with critical directions[END_REF]. We established in [START_REF]Existence and regularity for critical anisotropic equations with critical directions[END_REF] an existence result on cylindric domains and a general regularity result. We refer to Section 2, see Corollary 2.2, for examples of solutions of the limit problem (1.6). As a last remark concerning the case p + = p * , it is a borderline case for the boundedness of Palais-Smale sequences. Indeed, while any Palais-Smale sequence is bounded in case p + < p * for nonlinearities of the type |u| rα-2 u + f (x, u), r α → p * (see El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF]), we get examples of unbounded Palais-Smale sequences in case p + > p * for the nonlinearity |u| p + -2 u (see Fragalà-Gazzola-Kawohl [START_REF] Fragalà | Existence and nonexistence results for anisotropic quasilinear elliptic equations[END_REF]). Theorem 1.1 should be seen as a general key step in the analysis of blow-up for critical anisotropic equations. We have in view the numerous applications of bubble tree decompositions which have been developed for the equation -∆u = u 2 * -1 . Among other applications, bubble tree decompositions turn out to be key points in the use of topological arguments such as Lusternik-Schnirelmann equivariant categories. They also turn out to be key points in the analysis of ruling out bubbling and proving compactness of solutions. Possible references in book form on these subjects are Druet-Hebey-Robert [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF], Ghoussoub [START_REF] Ghoussoub | Duality and perturbation methods in critical point theory[END_REF], and Struwe [START_REF]Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems[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 [START_REF] Alves | Positive solutions to quasilinear equations involving critical exponent on perturbed annular domains[END_REF], 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], Guedda-Veron [START_REF] Guedda | Quasilinear elliptic equations involving critical Sobolev exponents[END_REF], Saintier [START_REF] Saintier | Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian[END_REF], and Yan [START_REF] Yan | A global compactness result for quasilinear elliptic equation involving critical Sobolev exponent[END_REF]. Needless to say, the above list does not pretend to exhaustivity.

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 possible 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], Boccardo-Gallouët-Marcellini [START_REF] Boccardo | Anisotropic equations in L 1[END_REF], Boccardo-Marcellini-Sbordone [START_REF] Boccardo | L ∞ -regularity for variational problems with sharp nonstandard growth conditions[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], Kolodīȋ [START_REF] Kolodīȋ | An estimate for the maximum modulus of generalized solutions of the Dirichlet problem for elliptic equations in divergence form[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], Marcellini [START_REF] Marcellini | Regularity and existence of solutions of elliptic equations with p, q-growth conditions[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]Existence and regularity for critical anisotropic equations with critical directions[END_REF][START_REF]Strong maximum principles for anisotropic elliptic and parabolic equations[END_REF].

We give examples of solutions and Palais-Smale sequences for problem (1.2) in Section 2, we discuss our geometric hypothesis on the domain Ω in Section 3, we prove a concentration result in Section 4, and we prove Theorem 1.1 in Section 5.

Examples of solutions and Palais-Smale sequences

This section is devoted to the construction of solutions and Palais-Smale sequences for problem (1.2). We begin with providing a class of examples in the situation where -→ p consists in two distinct exponents p -and p + . In other words, we assume in the following that there exist n -≥ 1 and

n + ≥ 1 such that n = n -+ n + , p 1 = • • • = p n -= p -, and p n -+1 = • • • = p n = p + . Proposition 2.
= • • • = p n -= p -and p n -+1 = • • • = p n = p + . Let ε and λ be two real numbers. Let Ω = Ω 1 × Ω 2 , where Ω 1 is a nonempty, open subset of R n -and Ω 2 is 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 + ε |v| p --2 v in Ω 1 , v ∈ D 1,p -(Ω 1 ) ∩ L p + (Ω 1 ) (∩L p -(Ω 1 ) if ε = 0 and Ω 1 is unbounded) , (2.1)
and let w be a solution in

D 1, - → p (Ω 2 ) of the problem      - n + i=1 ∂ ∂x i ∂w ∂x i p + -2 ∂w ∂x i = λ |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 by

u (x 1 , . . . , x n ) = v x 1 , . . . , x n -w x n -+1 , . . . , x n (2.3) is a solution in D 1, - → p (Ω) of the problem -∆ - → p u = λ |u| p + -2 u + ε |u| p --2 u in Ω , u ∈ D 1, - → p (Ω) ∩ L p + (Ω) (∩L p -(Ω) if ε = 0 and Ω is unbounded) . ( 2

.4)

Proof. A direct computation provides the result.

In case ε = 0, Ω 1 = R n -, and Ω 2 = R n + , see Vétois [START_REF]Existence and regularity for critical anisotropic equations with critical directions[END_REF], we have the following corollary.

Corollary 2.2. 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 problem (2.4) with ε = 0 and Ω = R n 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 ) = n -(n --p -) p --1 (p --1) p --1 n --p - p 2 - 1 1 + n - i=1 |x i | p - p --1 n --p - p - × W   n i=n -+1 |x i | p + p + -1 p + -1 p +   ,
where W is such that W > 0 and W ′ < 0 in (0, R), and W = 0 in [R, +∞) for some R > 0.

Since the function W has compact support, Corollary 2.2 provides a class of solutions of the purely invariant problem (1.6) on cylindric domains

Ω = R n -× V for all nonempty, open subsets V of R n + .
Now, we consider the case of a bounded domain Ω = Ω 1 × Ω 2 . Using a result by Franchi-Lanconelli-Serrin [START_REF] Franchi | Existence and uniqueness of nonnegative solutions of quasilinear equations in R n[END_REF], we get the existence of a solution of problem (2.2) which has compact support in Ω 2 for λ > 0 large (see Vétois [START_REF]Existence and regularity for critical anisotropic equations with critical directions[END_REF]). In order to apply Proposition 2.1, it remains to find solutions of problem (2.1).

In case n -≥ 3, p -= 2, and p + = 2 * , see Bahri-Coron [START_REF] Bahri | On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain[END_REF], we get the existence of a nonnegative, nontrivial solution of problem (2.1) with ε = 0 for noncontractible, smooth, bounded domains Ω 1 . Therefore, we state a second corollary of Proposition 2.1 as follows.

Corollary 2.3. Let n -≥ 3, n + ≥ 1, n = n -+ n + , and -→ p = (p 1 , . . . , p n ), and assume that

p 1 = • • • = p n -= 2, p n -+1 = • • • = p n = 2 * . Let Ω = Ω 1 × Ω 2 , where Ω 1 is a nonempty, noncontractible, smooth, bounded, open subset of R n -and Ω 2 is a nonempty, open subset of R n + . Given a point a = (a 1 , . . . , a n ) in Ω 2 , for λ > 0 large, there exists a nonnegative, nontrivial solution of the form (2.3), in D 1, - → p (Ω) ∩ C 1 (Ω), of problem (2.4) with ε = 0.
In case n -≥ 4, p -= 2, and p + = 2 * , see Brézis-Nirenberg [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], we get the existence of a nonnegative, nontrivial solution of problem (2.1) for ε > 0 small, on smooth, bounded domains Ω 1 . We then state a third corollary of Proposition 2.1 as follows.

Corollary 2.4. Let n -≥ 4, n + ≥ 1, n = n -+ n + , and -→ p = (p 1 , . . . , p n ), and assume that

p 1 = • • • = p n -= 2, p n -+1 = • • • = p n = 2 * . Let Ω = Ω 1 × Ω 2 , where Ω 1 is a nonempty, smooth, bounded, open subset of R n -and Ω 2 is a nonempty, open subset of R n + . Given a point a = (a 1 , . . . , a n ) in Ω 2 ,
for λ > 0 large and ε > 0 small, there exists a nonnegative, nontrivial solution of the form

(2.3), in D 1, - → p (Ω) ∩ C 1 (Ω), of the problem (2.4).
In case n -≥ 5, see Rey [START_REF] Rey | A multiplicity result for a variational problem with lack of compactness[END_REF][START_REF]The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF], we get the existence of families of solutions of problem (2.1) with 0 < ε ≪ 1 on smooth, bounded domains Ω 1 , which are bounded in D 1,2 (Ω 1 ) and which blow up as ε → 0 at nondegenerate, critical points of the regular part of Green's function. Using this result, we state a fourth corollary of Proposition 2.1 as follows.

Corollary 2.5. Let n -≥ 5, n + ≥ 1, n = n -+ n + , and -→ p = (p 1 , . . . , p n ), and assume that

p 1 = • • • = p n -= 2, p n -+1 = • • • = p n = 2 * . Let Ω = Ω 1 × Ω 2 ,
where Ω 1 is a nonempty, smooth, bounded, open subset of R n -and Ω 2 is a nonempty, open subset of R n + . Let (ε α ) α be a sequence of positive real numbers converging to 0. Given a point a = (a 1 , . . . , a n ) in Ω 2 , for λ > 0 large, there exists a bounded, blowing up, Palais-Smale sequence for the functional I ∞ defined in (1.10) with f ≡ 0.

We end this section with a more classical example of Palais-Smale sequences which holds in the general situation. We let n ≥ 3 and -→ p = (p 1 , . . . , p n ), and we assume that n i=1 1/p i > 1 and p + = p * . We let λ be a positive real number, Ω be a bounded, open subset of R n , and f be a Caratheodory function in Ω × R satisfying the growth condition (1.3). We let (r α ) α be a sequence of real numbers converging to p + , and we assume that q < r α < p + for all α, where q is the real number in (1.3). By standard variational arguments, for any α, we then get the existence of a function u α in D 1, - → p (Ω) such that there hold Ω |u α | rα dx = λ and

n i=1 1 p i Ω ∂u α ∂x i p i dx - Ω F (x, u α ) dx = min u∈D 1, - → p (Ω) Ω |u| rα dx=λ n i=1 1 p i Ω ∂u ∂x i p i dx - Ω F (x, u) dx ,
where F (x, u) = u 0 f (x, s) ds. In particular, for any α, there exists a positive real number λ α such that the function u α is a solution of problem (1.8). The sequence (u α ) α is a bounded Palais-Smale sequence for the functionals (I α ) α defined in (1.9).

The asymptotic stability of domains

This section is devoted to the geometric condition of asymptotic stability of domains which is in the statement of Theorem 1.1. The notion of asymptotically stable domains was introduced in El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF]. We state the definition below, and we comment this notion in our critical setting p + = p * . We first recall that a nonempty subset U of R n is said to satisfy the segment property if for any point a on ∂U , there exist a neighborhood X a of a and a nonzero vector σ a such that X a ∩ U + tσ a ⊂ U for all real numbers t in (0, 1). Adapting classical arguments, as developed, for instance, in Adams-Fournier [START_REF] Adams | Sobolev Spaces[END_REF], we get that any nonempty, open subset U of R n satisfying the segment property is such that the anisotropic Sobolev space D 1, - → p (U ) consists of the restrictions to U of functions in D 1, - → p (R n ) with support in U .

The definition of asymptotically stable domains states as follows.

Definition 3.1.

A nonempty, open subset Ω of R n is said to be asymptotically -→ p -stable if for any sequence (µ α ) α of positive real numbers converging to 0 and for any sequence

(x α ) α in R n , the sets Ω α = τ - → p µα,xα (Ω), where τ - → p
µα,xα is as in (1.5), converge, up to a subsequence, as α → +∞, to an open subset U of R n which is either empty or satisfies the segment property. The convergence is in the sense that the two following properties hold true:

(i) any compact subset of U is included in Ω α for α large, (ii) for any compact K ⊂ R n , there holds |K ∩ Ω α \U | → 0 as α → +∞. We use the notation Ω α → U when (i)-(ii) are satisfied.

Limits in the sense of (i)-(ii) are unique up to sets of measure zero. Uniqueness, without subtracting sets of measure zero, is recovered when requiring in addition that the limit set is open and satisfies the segment property. The asymptotic stability of domains is a subtle notion. Figure 2 below illustrates what can go wrong with a domain which is not asymptotically stable. The annular cylinder in Figure 2 bends on itself and converges to a domain of the type (R 2 \H) × [a, b], where a < b and H is a halfline. In particular, the limit domain does not satisfy the segment property, and thus the annular cylinder is not asymptotically stable. On the other hand, the ball, illustrated in Figure 1, provides an example of a domain which is asymptotically -→ p -stable for any anisotropic configuration -→ p . More generally we get the following result. Theorem 3.2. Let n ≥ 3 and -→ p = (p 1 , . . . , p n ), and assume that n i=1 1/p i > 1 and p + = p * . Then any nonempty, convex, bounded, open subset of R n is asymptotically -→ p -stable. Moreover, the limit domains U in Definition 3.1 can be chosen to be either empty or convex.

Proof. We refer to Vétois [67, Theorem 2.1] for the proof of Theorem 3.2 which works the same as in case p + < p * .

The convexity provides an easy geometric condition for asymptotic stability of domains. Contrary to the case p + < p * , there is no regularity type condition. Indeed, there exist domains which are not asymptotically stable for all anisotropic configurations satisfying p + = p * . For instance, we can easily show that annular domains of the type A = {x ∈ R n ; a < |x| < b}, where 0 < a < b, are not asymptotically stable in case p + = p * . In order to prove this claim, we let I + = i ∈ {1, . . . , n} ; p i = p + , i 0 be such that p i 0 = p + , x 0 = (x 0 1 , . . . , x 0 n ) be the point in R n such that x i 0 n = a and x i n = 0 for all i = i 0 , and (µ α ) α be a sequence of positive real numbers converging to 0. We then get that τ

- → p µα,x 0 (A) → x ∈ R n ; 0 < i∈I + x 2
i < b 2a 2 as α → +∞, and that the limit domain does not satisfy the segment property. It follows that the domain A, even though being smooth, is not asymptotically stable in case p + = p * . In Theorem 3.3 below, we give a general class of asymptotically stable domains as products of two domains, one of which being asymptotically stable with respect to noncritical directions, and the other satisfying the segment property in critical directions. One can find a detailed discussion in El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF] and Vétois [START_REF]Asymptotic stability, convexity, and Lipschitz regularity of domains in the anisotropic regime[END_REF] about asymptotically stable domains with respect to noncritical directions. Our result states as follows.

Theorem 3.3. Let n ≥ 3, -→ 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 Ω 1 be a nonempty, bounded, open subset of R n-n + and Ω 2 be a nonempty, bounded, open subset of R n + . Then Ω 1 × Ω 2 is asymptotically -→ p -stable if and only if Ω 1 is asymptotically -→ q -stable, where -→ q = (p 1 , . . . , p n-n + ), and Ω 2 satisfies the segment property.

Proof of Theorem 3.3. We let (µ α ) α be a sequence of positive real numbers converging to 0, (x 1 α ) α be a sequence of points in R n-n + , and (x 2 α ) α be a sequence of points in R n + . For any α, we let x α = (x 1 α , x 2 α ). We remark that τ

- → p µα,xα (Ω 1 × Ω 2 ) = τ - → q µα,x 1 α (Ω 1 ) × Ω 2 -x 2 α , (3.1)
where -→ p = (p 1 , . . . , p n ) and -→ q = (p 1 , . . . , p n-n + ). We first assume that Ω 1 is asymptotically -→ q -stable, and that Ω 2 satisfies the segment property. Passing if necessary to a subsequence, we get that there exists an open subset U 1 of R n-n + , either empty or satisfying the segment property, such that τ

- → q µα,x 1 α (Ω 1 ) → U 1 as α → +∞ in the sense of Definition 3.1. In case |x 2 α | → +∞ as α → +∞, we easily get that τ - → p µα,xα (Ω 1 × Ω 2 ) → ∅ as α → +∞.
Therefore, passing if necessary to another subsequence, we may assume that the sequence (x 2 α ) α converges to a point a 2 in R n + . We then claim that

Ω 2 -x 2 α → Ω 2 -a 2 as α → +∞, or equivalently that Ω 2 -z α → Ω 2 as α → +∞, where z α = x 2 α -a 2 . For any compact subset K of Ω 2 , we get d (K, R n \ (Ω 2 -z α )) ≥ d (K, R n \Ω 2 ) -|z α | for all α. Since z α → 0 as α → +∞, it follows that lim inf α→+∞ d (K, R n \ (Ω 2 -z α )) > 0,
and thus that K ⊂ Ω 2z α for α large. Now, we prove that for any compact subset K of R n + , there holds |K ∩ (Ω 2z α ) \Ω 2 | → 0 as α → +∞. As is easily seen, it suffices to show that for any point a in R n + , there exists an open neighborhood X a of a such that |X a ∩ (Ω 2z α ) \Ω 2 | → 0 as α → +∞. In case a belongs to Ω 2 , for r > 0 small, there holds B a (r) ⊂ Ω 2 , and thus |B a (r) ∩ (Ω 2z α ) \Ω 2 | = 0 for all α. In case a belongs to R n + \Ω 2 , for r > 0 small, there holds B a (r

) ⊂ R n + \Ω 2 , B a (r) ⊂ R n + \ (Ω 2 -z α ) for α large,
and thus |B a (r) ∩ (Ω 2z α ) \Ω 2 | = 0 for α large. Now, we assume that the point a belongs to the boundary of Ω 2 . Since Ω 2 satisfies the segment property, by Maz ′ ya-Poborchi [49, Section 1.3.1], we get that there exists a Cartesian coordinate system ξ 1 , . . . , ξ n + of R n + , a continuous function ϕ a : R n + -1 → R, and an open neighborhood X a of a such that the set X a ∩ Ω 2 consists of the points ξ 1 , . . . , ξ n + in X a satisfying ξ n + < ϕ a ξ 1 , . . . , ξ n + -1 . We then let Y a be a bounded, open neighborhood of a such that Y a ⊂ X a . Since z α → 0 as α → +∞, it follows that, for α large, the set

Y a ∩ (Ω 2 -z α ) \Ω 2 consists of the points ξ 1 , . . . , ξ n + in Y a satisfying ϕ a ξ 1 , . . . , ξ n + -1 ≤ ξ n + < ϕ a ξ 1 + z α 1 , . . . , ξ n + -1 + z α n + -1 -z α n +
, where z α = z α 1 , . . . , z α n + in the coordinate system ξ 1 , . . . , ξ n + . By the continuity of the function ϕ a , we then get |Y a ∩ (Ω 2z α ) \Ω 2 | → 0 as α → +∞. This ends the proof of our claim, namely that

Ω 2 -x 2 α → Ω 2 -a 2 as α → +∞. Taking into account (3.1), we then get τ - → p µα,xα (Ω 1 × Ω 2 ) → U 1 × (Ω 2 -a 2 ) as α → +∞. Moreover, in case U 1 is nonempty, since both U 1
and Ω 2 satisfy the segment property, we get that U 1 × (Ω 2a 2 ) satisfies the segment property. Since this holds true for all sequences (µ α ) α and (x α ) α , it follows that Ω 1 is asymptotically -→ p -stable. Conversely, we assume that Ω 1 ×Ω 2 is asymptotically -→ p -stable. For any α, we let x α = (x 1 α , x 2 α ), where x 2 α = 0 is fixed. Passing if necessary to a subsequence, we get that there exists an open subset U of R n , either empty or satisfying the segment property, such that τ

- → p µα,xα (Ω 1 × Ω 2 ) → U as α → +∞. We let U 1 = {x ∈ R n-n + ; ∃y ∈ Ω 2 (x, y) ∈ U }. We claim that U = U 1 × Ω 2 . Clearly, we get U ⊂ U 1 × Ω 2 .
On the other hand, we proceed by contradiction and assume that there exists a point (x 1 , x 2 ) in U 1 ×Ω 2 which does not belong to U . By definition of U 1 , there exists a point y 2 in Ω 2 such that (x 1 , y 2 ) belongs to U . It follows that B x 1 (ε) × B y 2 (ε) ⊂ U for small positive real numbers ε. Moreover, since τ

- → p µα,xα (Ω) → U , we then get B x 1 (ε) ⊂ τ - → q µα,x 1
α (Ω 1 ) for α large, and we also get |(B x 1 (ε) × B x 2 (ε)) \U | = 0. This is in contradiction with the segment property satisfied by U . This ends the proof of our claim, namely that U = U 1 × Ω 2 . By (3.1) and since τ -→ p µα,xα (Ω) → U , it follows that τ -→ q µα,x 1 α (Ω 1 ) → U 1 as α → +∞. Moreover, since U satisfies the segment property, we also get that both U 1 and Ω 2 satisfy the segment property. Since this holds true for all sequences (µ α ) α and (x 1 α ) α , it follows that Ω 1 is asymptotically -→ q -stable. This ends the proof of Theorem 3.3. In Figure 3, we rescale the cylinder B 2 × (0, 1), where B 2 is the unit two-dimensional ball. The limit domain is of the form H × (0, 1), where H is a halfplane. In Figure 4, we rescale the cylinder (0, 1) 2 × B 2 (in projection onto {0} × R 3 ). The limit domain is R 2 × B 2 .

The concentration on critical directions

In this section, we prove a concentration result which is the main tool in the proof of Theorem 1.1. We let (Ω α ) α be a sequence of nonempty, bounded, open subsets of R n , (λ α ) α be a sequence of positive real numbers, and (r α ) α be a sequence of real numbers in (1, p + ] converging to p + . For any α, we define the functional (J α ) α by

J α (u) = n i=1 1 p i Ωα ∂u ∂x i p i dx - λ α r α Ωα |u| rα dx (4.1)
for all functions u in D 1, - → p (Ω α ). We say that a sequence (u α ) α is Palais-Smale for the function-

als (J α ) α if there hold u α ∈ D 1, - → p (Ω α ), |J α (u α )| ≤ C for some positive constant C independent of α, and DJ α (u α ) D 1, - → p (Ωα) ′ → 0 as α → +∞. Moreover, (u α ) α is said to be bounded if there hold λ α Ωα |u α | rα dx ≤ C and λ p + p + -1 α Ωα |u α | p + (rα-1) p + -1 dx ≤ C (4.2)
for some positive constant C independent of α. In particular, bounded Palais-Smale sequences for the functionals (J α ) α are bounded in D 1, - → p (R n ). Our concentration result states as follows.

Proposition 4.1. Let n ≥ 3 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 ≤ nn + . Let U be an open subset of R n either empty or satisfying the segment property, and (Ω α ) α be a sequence of nonempty, bounded, open subsets of R n converging to U in the sense of Definition 3.1. Assume that there exists a bounded, open subset V of R n + such that the sets U and Ω α are included in R n-n + × V for all α. Let (λ α ) α be a sequence of positive real numbers converging to λ ≥ 0, and (r α ) α be a sequence of real numbers in (1, p + ] converging to p + . Let (u α ) α be a bounded Palais-Smale sequence for the functionals (J α ) α defined in (4.1). Then there exists a function u ∞ in D 1, - → p (U ) (u ∞ = 0 in case U = ∅) and a finite number k of distinct points y j = y j 1 , . . . , y j n-n + in R n-n + , j = 1, . . . , k, and nonnegative, nontrivial, finite measures ξ j with support in V y j , j = 1, . . . , k, where

V y j = y j 1 , . . . , y j n-n + × V (4.
3) such that, up to a subsequence, there holds

|u α | p + -⇀ |u ∞ | p + + k j=1 ξ j (4.4)
as α → +∞ in the sense of measures on compact subsets of R n . Moreover, there exists a positive constant Λ 0 = Λ 0 ( -→ p ) such that for any j = 1, . . . , k, there holds

λ n+p + p + ξ j V y j ≥ Λ 0 , (4.5)
where V y j is as in (4.3). In particular, if λ = 0, then k = 0.

Proof of Proposition 4.1. Passing if necessary to a subsequence, we may assume that the sequence (u α ) α converges weakly to a function u ∞ in D 1, - → p (R n ) and that (u α ) α converges to u ∞ almost everywhere in R n . Since Ω α → U as α → +∞ in the sense of Definition 3.1, we get that the support of the function u ∞ is included in U . In case U = ∅, since U satisfies the segment property, it follows that the function u ∞ belongs to the anisotropic Sobolev space D 1, - → p (U ). We fix a positive real number R, and we let B 0 (R) be the (nn + )-dimensional ball of center 0 and radius R. For any α, we let v α = u αu ∞ . By Banach-Alaoglu theorem, since the sequence (v α ) α is bounded in D as α → +∞ for all bounded, measurable functions ϕ on B 0 (2R) × R n + . Borrowing 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 , see the analysis in Vétois [START_REF]Existence and regularity for critical anisotropic equations with critical directions[END_REF], we can prove that there exists an at most countable index set J of distinct points y j in B 0 (R), j ∈ J, such that

Supp µ ∩ B 0 (R) × V ⊂ j∈J V y j , (4.8)
where V y j is as in (4.3). Moreover, 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 that for any Borelian set A in B 0 (R), there holds

µ A × V ≤ Λ n i=1 ν i A × V p + np i (4.9)
for some positive constant Λ = Λ ( -→ p ). We then prove (4.

u α L p + (Ωα) ∂ϕ ∂x i L ∞ (R n ) . (4.12)
By (4.6), (4.7), (4.11), and (4.12), passing to the limit into (4.10) as α → +∞, we get

n i=1 B 0 (2R)×V ϕdν i - n i=1 U ∂u ∞ ∂x i p i ϕdx + n-n + i=1 U ψ i u ∞ ∂ϕ ∂x i dx ≤ λ U |u ∞ | p + ϕdx + B 0 (2R)×V ϕdµ + R n ϕdx + C n i=n-n + +1 ∂ϕ ∂x i L ∞ (R n ) (4.13)
for some positive constant C independent of ϕ. We claim that

n i=1 U ψ i u ∞ ∂ϕ ∂x i dx + n i=1 U ψ i ∂u ∞ ∂x i ϕdx = λ U |u ∞ | p + ϕdx . ( 4
∂u α ∂x i p i -2 ∂u α ∂x i b α ∂ϕ ∂x i dx + n i=1 Ωα ∂u α ∂x i p i -2 ∂u α ∂x i ∂b α ∂x i ϕdx = λ α Ωα |u α | rα-2 u α b α ϕdx + DJ α (u α ) . (b α ϕ) = λ α Ωα |u α | rα-2 u α b α ϕdx + o (1) (4.15)
as α → +∞. By (4.2), we get that the functions

λ α |u α | rα-2 u α keep bounded in L p + /(p + -1) (R n ).
Moreover the functions λ α |u α | rα-2 u α converge, up to a subsequence, almost everywhere to

λ |u ∞ | p + -2 u ∞ in R n as α → +∞.
By standard integration theory, it follows that the functions

λ α |u α | rα-2 u α converge weakly to λ |u ∞ | p + -2 u ∞ in L p + /(p + -1) (R n ).
ϕdν i - n i=1 U ∂u ∞ ∂x i p i ϕdx - n i=1 U ψ i ∂u ∞ ∂x i ϕdx ≤ λ B 0 (2R)×V ϕdµ + R n ϕdx + C n i=n-n + +1 ∂ϕ ∂x i L ∞ (R n ) . (4.16)
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,σ,y into (4.16), and passing to the limit as ε → 0, we get

n i=1 ν i V y ≤ λµ V y , (4.17)
where V y is as in (4.3). By (4.9) and (4.17), we get that there holds either

µ V y = 0 or λµ V y p + n+p + ≥ Λ -n n+p + (4.18)
for all points y in R n-n + . In particular, since the measure µ is finite, it follows from (4.18) that the index set J in (4.8) is finite. Passing to the limit as R → +∞, by (4.6) and (4.8), we get the decomposition (4.4). Finally, (4.5) follows from (4.18).

Proof of the main result

In this section, we prove Theorem 1.1 by using Proposition 4.1 and Proposition 5.1 below. Proposition 5.1. Let n ≥ 3 and -→ p = (p 1 , . . . , p n ), and assume that n i=1 1/p i > 1 and p + = p * . Then there exists a positive constant E 0 = E 0 ( -→ p ) such that for any positive real number λ and any -→ p -bubble (B α ) α of multiplier λ, there holds

E (B α ) ≥ E 0 λ -n/p + , (5.1)
where E (B α ) is as in (1.7).

Proof of Proposition 5.1. 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 ≤ nn + . We let λ be a positive real number and (B α ) α be a -→ p -bubble of profile u and multiplier λ. 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 .

We then easily get (5.1).

Proof of Theorem 1.1. 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 ≤ nn + . We let (u α ) α be a bounded Palais-Smale sequence for the functionals (I α ) α defined in (1.9). Passing if necessary to a subsequence, we may assume that (u α ) α converges weakly to a function u ∞ in D 1, - → p (Ω).

Proceeding in the same way as in El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF], we can prove that u ∞ is a solution of problem (1.2). By an easy adaptation of the argument in Brézis-Lieb [START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF] and by the growth condition (1.3), we also get that, up to a subsequence, (u 0 α ) α , where u 0 α = u αu ∞ , is Palais-Smale for the functionals (J α ) α defined in (4.1) with λ α = λ and Ω α = Ω for all α, and that there holds J α (u 0 α ) = I α (u α ) -I α (u ∞ ) + o (1) as α → +∞. If the sequence (u α ) α converges strongly, up to a subsequence, to the function u ∞ in D 1, - → p (Ω), then we get (1.11) and (1.12)

with k = 0. From now on, subtracting if necessary the weak limit u ∞ , we may assume that the sequence (u α ) α converges weakly but not strongly to 0 in D 1, - → p (Ω), and that (u α ) α is Palais-Smale for the functionals (J α ) α with λ α = λ and Ω α = Ω for all α. In particular, passing if necessary to a subsequence, we get lim inf for any positive real number s and for any point y = (y 1 , . . . , y n ) in R n . By (5.4) and since the domain Ω is bounded, we get the existence of two positive real numbers s 0 and δ 0 such that there holds Q α (s 0 ) > δ 0 for all α. By the continuity of the functions Q α , it follows that there exists a sequence (µ α ) α of real numbers in (0, s 0 ) 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 max

  : R n → R n by τ -→ p µ,a (x 1 , . . . , x n ) = µ p 1 -p * p 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.

Figure 2 .

 2 Figure 2. Rescaling of an annular cylinder (n = 3, p 1 = 1.1, p 2 = 2, p 3 = 44/9). The first line describes the scale in the rescaling. The second line describes the deformation of the domain.

Figure 3 .

 3 Figure 3. Rescaling of a cylinder (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.

Figure 4 .

 4 Figure 4. Rescaling of (0, 1) 2 × B 2 (n = 4, p 1 = p 2 = 1.5, p 3 = p 4 = 6, projection onto {0} × R 3 ). The first line describes the scale in the rescaling. The second line describes the deformation of the domain.

Figures 3 and 4

 4 Figures3 and 4above illustrate Theorem 3.3 in the cases of two different cylindric domains. In Figure3, we rescale the cylinder B 2 × (0, 1), where B 2 is the unit two-dimensional ball. The limit domain is of the form H × (0, 1), where H is a halfplane. In Figure4, we rescale the cylinder (0, 1) 2 × B 2 (in projection onto {0} × R 3 ). The limit domain is R 2 × B 2 .

. 14 )

 14 Since the function u ∞ belongs to D 1, - → p (U ) and since Ω α → U as α → +∞ in the sense of Definition 3.1, we get that there exist smooth functions b α with compact support in Ω α converging to u ∞ as α → +∞ in D 1, - → p (R n ), and thus also inL p + (R n ) by the continuity of the embedding of D 1, - → p (R n ) into L p + (R n ).Since the sequence (u α ) α is Palais-Smale for the functionals (J α ) α defined in (4.1), it follows that n i=1 Ωα

Ω

  By Palais-Smale properties and by Hölder's inequality, it follows that|u α | rα dx + o (1) ≤ |Ω| p + -rα p + Ω |u α | p + dx rα p + + o (1)as α → +∞, and thus that lim infα→+∞ Ω|u α | p + dx > 0 .(5.4)For any α, we then define the concentration functionQ α : R + → R + by Q α (s) = max y∈Ω P ) = (x 1 , . . . , x n ) ∈ R n ; |x iy i | < s p + -p i p i∀i ∈ {1, . . . , nn + }

  )|u α | p + dx = P -→ p xα (µα) |u α | p + dx = δ 0 (5.5)

  1 below is the basic tool in our constructions. We state our examples in Corollaries 2.2, 2.3, 2.4, and 2.5. Proposition 2.1 relies on a direct computation.

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

  1, - → p (U ) and since the sets U and Ω α are included in 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 . Since the sequence (u α ) α is bounded in L p + (R n ) and converges to u ∞ almost everywhere in R n , by Brezis-Lieb[START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF], we get +∞ for all bounded, measurable functions ϕ on B 0 (2R) × R n + . Since there holds|∂u α /∂x i | p i ≥ |∂v α /∂x i | p i -|∂u ∞ /∂x i | p i, where v α = u αu ∞ for all α and i = 1, . . . , n, we

	U	u p + α ϕdx -→	U	u p + ∞ ϕdx +	B 0 (2R)×V	ϕdµ		(4.6)
	as α → also get
	lim inf α→+∞ U	∂u α ∂x i	p i	ϕdx ≥		B 0 (2R)×V	ϕdν i -	U	∂u ∞ ∂x i	p i	ϕdx	(4.7)

  5) by using Palais-Smale properties of the sequence (u α ) α . For any nonnegative, smooth function ϕ with support in B 0 (2R)×R n + , we get +∞. 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) (R n ), and thus converge, up to a subsequence, weakly to some function ψ i in L p i /(p i -1) (R n ) as α → +∞. Moreover, for any i = 1, . . . , nn + , since p i < p + , 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 +∞. For any α and any i = nn + + 1, . . . , n, we get

	n i=1 as α → Ωα = λ α Ωα ≤ λ α ∂u α ∂u α ∂x i Ωα |u α | rα ϕdx + DJ α (u α ) . (u α ϕ) p i ϕdx + Ωα ∂u α ∂x i p i -2 ∂u α ∂x i u α Ωα |u α | p + ϕdx + Ωα ϕdx + o (1) ∂ϕ ∂x i ∂x i p i -2 ∂u α ∂x i u α ∂ϕ ∂x i dx -→ U ψ i u ∞ ∂ϕ dx dx ∂x i as α → Ωα ∂u α ∂x i p + -2 ∂u α ∂x i u α ∂ϕ ∂x i p + -1 ∂u α dx ≤ ∂x i L p + (Ωα)	(4.10) (4.11)

  for some positive constant Λ independent of u. By Young's inequality, it follows that for any positive real number ε, there holds

	Taking ε = n Λλ , since n-n + i=1	1 p i = n-n + +p + p +	, it follows from (5.2) and (5.3) that
		n-n + i=1	R n	∂u ∂x i	p i	dx ≤	Λλ n	n n-n +	(n -n + )	n-n + i=1	R n	∂x i ∂u	p i	dx	n-n + +p + n-n +
							R n	|u| p + dx ≤ Λ	n i=1	R n	∂u ∂x i	p i	dx	p + np i
	R n	|u| p + dx ≤	Λ n	(n -n + ) ε	-n + n-n +	n-n + i=1			R n	∂u ∂x i	p i	dx	p + (n-n + )p i
															+ ε	n i=n-n + +1 R n	∂u ∂x i	p +	dx . (5.2)
	On the other hand, since u is a solution of problem (1.6), we get
								n i=1 R n	∂u ∂x i	p i	dx = λ	R n	|u| p + dx .	(5.3)

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.

for all α. We claim that if the constant δ 0 is chosen small enough, then the sequence (µ α ) α converges to 0. Indeed, if not the case, then for any ε > 0, there exists s ε > 0 such that for any α and any point y in Ω, there holds

(5.6) By Proposition 4.1 with λ α = λ and Ω α = Ω for all α (in this case, we can easily drop the segment condition), it follows from (5.6) that the sequence (u α ) α converges in fact strongly to 0 in L p + (Ω). There is a contradiction with (5.4), and this proves our claim, namely that if the constant δ 0 is chosen small enough, then the sequence (µ α ) α converges to 0. For any α, we then define the function u α by

where τ -→ p µα,xα is as in (1.5). As well as (u α ) α , the sequence ( u α ) α is Palais-Smale for the functionals (J α ) α with λ α = λµ p + -rα α and Ω α = τ -→ p µα,xα (Ω) for all α. Passing if necessary to a subsequence, we may assume that the sequence (λ α ) α converges to a nonnegative real number λ. By an easy change of variable, we also find

and

It follows from (5.7) and (5.8) that ( u α ) α is a bounded Palais-Smale sequence for the functionals (J α ) α . Passing if necessary to a subsequence, since the domain Ω is asymptotically -→ p -stable, we may assume that there exists an open subset U , either empty or satisfying the segment property, such that Ω α → U as α → +∞ in the sense of Definition 3.1. Moreover, since Ω is bounded and since the maps τ -→ p µα,xα do not rescale the domain in critical directions, we get that the sets Ω α and U are included in R n-n + × V for some bounded, open subset V independent of α. By Proposition 4.1, we get the existence of a function u in D 1, - → p (U ) ( u = 0 in case U = ∅) and a finite number k of distinct points y j = y j 1 , . . . , y j n-n + in R n-n + , j = 1, . . . , k, and nonnegative finite measures ξ j with support in V y j , j = 1, . . . , k, such that, up to a subsequence, there holds

as α → +∞ in the sense of measures on compact subsets of R n . If λ = 0, then k = 0. Otherwise, in case λ > 0, we claim that if the constant δ 0 is chosen small enough, then k = 0. We assume by contradiction that k = 0. By (5.5) and by an easy change of variable, for any α and j = 1, . . . , k, we get

(5.9)

Since there holds V y j ⊂ P -→ p y j (1), passing to the limit into (5.9) as α → +∞, it follows that

| u| p + dx + ξ j V y j ≤ δ 0 .

(5.10)

On the other hand, by (4.5), we get ξ j V y j ≥ Λ 0 λ -(n+p + )/p + . Since there hold µ p + -rα α ≤ 1 for α large and λµ p + -rα α → λ as α → +∞, we get λ ≤ λ, and thus ξ j V y j ≥ Λ 0 λ -(n+p + )/p + . There is a contradiction with (5.10) in case δ 0 < Λ 0 λ -(n+p + )/p + . This ends the proof of our claim, namely that if the constant δ 0 is chosen small enough, then k = 0. By Radon's theorem, it follows that the sequence ( u α ) α converges to u in L p + loc (R n ). By (5.5) and by an easy change of variable, for any α, we get

(5.11)

Passing to the limit into (5.11) as α → +∞, it follows that

| u| p + dx = δ 0 , and thus that the function u is not identically zero. In particular, the domain U is not empty. The end of the proof follows Struwe [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF] (see El Hamidi-Vétois [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF] for the case of the anisotropic Laplace operator with p + < p * ), and we sketch it for sake of completeness. We prove that the function u is a solution of problem (1.6) with λ = λ, so that we can define the

We then prove that there exists a sequence (R 1 α ) α converging strongly to 0 in D 1, - → p (R n ) such that, up to a subsequence, (u 1 α ) α , where

as α → +∞. We iterate the above arguments in order to construct the -→ pbubbles (B 1 α ) α , . . . , B k α α . Our induction has to stop for some finite number k since, by Proposition 5.1, the energy of -→ p -bubbles is bounded from below by a positive constant which only depends on λ and on the anisotropic configuration -→ p . We finally obtain (1.11) and (1.12). Moreover, in case the functions u α are nonnegative, their weak limit u ∞ is nonnegative, and by considering the functions max (u j α -B j α , 0), we prove that the -→ p -bubbles (B 1 α ) α , . . . , B k α α are also nonnegative (see [START_REF] Hamidi | Sharp Sobolev asymptotics for critical anisotropic equations[END_REF]). This ends the proof of Theorem 1.1.