THE BLOW UP OF CRITICAL ANISTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

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THE BLOW-UP OF CRITICAL ANISTROPIC EQUATIONS WITH CRITICAL DIRECTIONS JEROME VETOIS Abstract. We investigate blow-up theory for doubly critical anisotropic problems in bounded domains of the Euclidean space. 1. Introduction Anisotropic operators modelize directionally dependent phenomena. In this paper, we con- sider problems posed on domains in the Euclidean space Rn in dimension n ≥ 2, and we provide ourselves with an anisotropic configuration ??p = (p1, . . . , pn) with pi > 1 for all i = 1, . . . , n. We define the anisotropic Laplace operator ∆??p by ∆??p u = n∑ i=1 ∂ ∂xi ?pixiu , (1.1) where ?pixiu = |∂u/∂xi| pi?2 ∂u/∂xi 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 [11] and Bendahmane–Langlais–Saad [13], as a model describing the spread of an epidemic disease in heterogeneous environments. They also emerge, see Antontsev–Dıaz–Shmarev [5] and Bear [10], 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 = ? |u| p??2 u+ f (·, u) in ? , u ? D1, ??p (?) , (1.2) where ∆??p is as in (1.1), ?

• let ?


• direction being


• tree decompositions


• configuration ??p


• bubble tree


• critical directions


• ??p


• anisotropic problems


• critical anisotropic

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• Publié par: pefav
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 1 Mo

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THE

BLOW-UP OF CRITICAL ANISTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

´ ˆ ´ JEROME VETOIS

Abstract.for doubly critical anisotropic problems in boundedWe investigate blow-up theory domains of the Euclidean space.

1.Introduction

Anisotropic operators modelize directionally dependent phenomena. In this paper, we con- sider problems posed on domains in the Euclidean spaceR n in dimensionn≥2, and we provide ourselves with an anisotropic configuration − p → = (p 1 , . . . , p n ) withp i >1 for alli= 1, . . . , n. We define the anisotropic Laplace operator Δ −→ p by n ∂ Δ −→ p u= X 1 ∂x i r x p i i u ,(1.1) i= wherer xp i i u=|∂u/∂x i | p i −2 ∂u/∂x i for alli= 1, . . . , n. Nonlinear equations of the type Δ − p → u=f(∙, u appear, for instance, in biology, They) appear in several places in the literature. see Bendahmane–Karlsen [11] and Bendahmane–Langlais–Saad [13], as a model describing the spread of an epidemic disease in heterogeneous environments. They also emerge, see Antontsev–D´ıaz–Shmarev [5] and Bear [10], 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 Δ − p → u=λ|u| p ∗ −2 u+f(∙, u) inΩ , ( u∈ − D 1, −→ p (Ω), (1.2) where Δ − p → is as in (1.1),Ωis a domain ofR 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 inΩwith respect to the normkuk D 1, − p → ( = P n k∂u/∂x i k L p i (Ω) ,p ∗ is the critical Sobolev Ω)i=1 exponent (see (1.4) below),λis a positive real number, andfis a Caratheodory function in Ω×Rsatisfying the growth condition |f(∙, u)| ≤C  |u| q−1 + 1  a.e. inΩ(1.3) for some real numberqin (1, p ∗ ) and for some positive constantCindependent ofu. We are concerned with the doubly critical situationp + =p ∗ , wherep +  (= maxp 1 , . . . , p n ) is the maximum value of the anisotropic configuration andp ∗ is as above the critical Sobolev exponent for the embeddings of the anisotropic Sobolev spaceD 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

Date:February 25, 2010.Revised:September 18, 2010. Published inNoDEA Nonlinear Differential Equations and Applications18(2011), no. 2, 173–197. 1

CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

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operator or thep-Laplace operator. Giveni= 1, . . . , n, thei-th direction is said to be critical ifp i =p ∗  if subcritical, resp.p i < p ∗  directions induce a failure in the rescaling. Critical invariance rule associated with (1.2). Given an anisotropic configuration − p → satisfying P n i=1 1/p i >1 andp j ≤n/  P n i=1 p 1 i −1  for allj= 1, . . . , n, the critical Sobolev exponent is equal to p ∗ = P n n p 1 i −1 .(1.4) i=1 Possible references on anisotropic Sobolev spaces are Besov [14], Haˇskovec–Schmeiser [38], Kruzhkov–Kolodı¯ı˘[40],Kruzhkov–Korolev[41],Lu[47],Nikol 0 sokakı´n,45[,]55dan,]´R[ı35ks˘i Troisi [65]. 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 anyµ >0 and any pointa= (a 1 , . . . , a n ) inR n , we define the affine transformation τ µ − p → ,a :R n →R n by ∗ − τ µ −→ p a, (x 1 , . . . , x n ) =  µ p 1 p − 1 p ∗ (x 1 −a 1 ), . . . , µ n p p n p (x n −a n )  .(1.5) As is easily checked, (1.5) provides a general rescaling invariance rule associated with problem ∗ (1.2) withf≡ particular,0. Inusolves−Δ − p → u=|u| p−2 uinΩif and only if the function v=µ −1 u◦  τ µ −→ p ,a  −1 solves−Δ − p → v=|v| p ∗ −2 vinτ µ − p → ,a (Ω), where ∗  τ µ − p → ,a  −1 (x 1 , . . . , x n ) =  µ p ∗ p − 1 p 1 x 1 +a 1 , . . p− n p +a n  . . , µ n p x n Given (µ α ) α of positive real numbers converging to 0, (a sequence x α ) α a converging sequence inR n λa positive real number,Ua nonempty, open subset ofR n , andua nontrivial solution , , inD 1 −→ p (U) of the problem =λ ( u − ∈ Δ D −→ p 1 u , − p → (U) |u , | p ∗ −2 uinU , (1.6) where Δ −→ p is as in (1.1), we call − p → -bubbleof centers (x α ) α , weights (µ α ) α , multiplierλ, domain U, and profileu, the sequence (B α ) α defined by 1 −→ B α =u◦τ µ p α ,x α µ α µ −→ p α ,x α  lts can find existence and regularity res Oneis as in (1.5). for problem for allα, whereτu (1.6)inV´etois[68].Inthefollowing,weimplicitlyextendprofilesofbubblesby0outsideof their domains so as to regard them as functions inD 1, − p → (R n ). With the above notations, we define the energyE(B α ) of a − p → -bubble (B α ) α by n E(B α ) = X 1 i=1 p i Z R n  ∂u ∂x i  p i dx− p λ ∗ Z R n |u| p ∗ dx= i= n X 1 p ∗ p ∗ − p i p i Z R n  u ∂x ∂ i  p i dx ,(1.7) the second equality in (1.7) being obtained by testing (1.6) withuand integrating by parts. We approximate problem (1.2) with the problems ( u − ∈ Δ D −→ p 1 u , −→ p ( =λ Ω α ) | , u| r α −2 u+f(∙, u) inΩ , (1.8)