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THE BLOW UP OF CRITICAL ANISTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

20 pages
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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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 spaceRnin dimensionn2, and we provide ourselves with an anisotropic configurationp= (p1, . . . , pn) withpi>1 for alli= 1, . . . , n. We define the anisotropic Laplace operator Δ−→pby nΔ−→pu=X1∂xirxpiiu ,(1.1) i= whererxpiiu=|∂u/∂xi|pi2∂u/∂xifor alli= 1, . . . , n. Nonlinear equations of the type Δpu=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 Δpu=λ|u|p2u+f(, u) inΩ , (uD1,−→p(Ω),(1.2) where Δpis as in (1.1),Ωis a domain ofRn,D1,−→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 normkukD1,p(=Pnk∂u/∂xikLpi(Ω),pis 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|q1+ 1a.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+ (= maxp1, . . . , pn) is the maximum value of the anisotropic configuration andpis as above the critical Sobolev exponent for the embeddings of the anisotropic Sobolev spaceD1,−→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 ifpi=p if subcritical, resp.pi< p directions induce a failure in the rescaling. Critical invariance rule associated with (1.2). Given an anisotropic configurationpsatisfyingPni=11/pi>1 andpjn/ Pni=1p1i1for allj= 1, . . . , n, the critical Sobolev exponent is equal to p=Pnnp1i1.(1.4) i=1 Possible references on anisotropic Sobolev spaces are Besov [14], Haˇskovec–Schmeiser [38], KruzhkovKolodı¯ı˘[40],KruzhkovKorolev[41],Lu[47],Nikol0sokakı´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= (a1, . . . , an) inRn, we define the affine transformation τµp,a:RnRnby τµ−→pa,(x1, . . . , xn) =µp1p1p(x1a1), . . . , µnppnp(xnan).(1.5) As is easily checked, (1.5) provides a general rescaling invariance rule associated with problem (1.2) withf particular,0. InusolvesΔpu=|u|p2uinΩif and only if the function v=µ1uτµ−→p,a1solvesΔpv=|v|p2vinτµp,a(Ω), where τµp,a1(x1, . . . , xn) =µpp1p1x1+a1, . .pnp+an. . , µnpxn Given (µα)αof positive real numbers converging to 0, (a sequence xα)αa converging sequence inRnλa positive real number,Ua nonempty, open subset ofRn, andua nontrivial solution , , inD1−→p(U) of the problem =λ (uΔD−→p1u,p(U)|u,|p2uinU ,(1.6) where Δ−→pis as in (1.1), we callp-bubbleof centers (xα)α, weights (µα)α, multiplierλ, domain U, and profileu, the sequence (Bα)αdefined by 1−→ Bα=uτµpα,xα µα µ−→pα,xα lts can find existence and regularity res Oneis as in (1.5). for problem for allα, whereτu (1.6)inV´etois[68].Inthefollowing,weimplicitlyextendprolesofbubblesby0outsideof their domains so as to regard them as functions inD1,p(Rn). With the above notations, we define the energyE(Bα) of ap-bubble (Bα)αby n E(Bα) =X1 i=1piZRnuxipidxpλZRn|u|pdx=i=nX1pppipiZRnuxipidx ,(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 (uΔD−→p1u,−→p(=λΩα)|,u|rα2u+f(, u) inΩ ,(1.8)
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