A PRIORI ESTIMATES FOR SOLUTIONS OF ANISOTROPIC ELLIPTIC EQUATIONS

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A PRIORI ESTIMATES FOR SOLUTIONS OF ANISOTROPIC ELLIPTIC EQUATIONS JEROME VETOIS Abstract. We prove universal, pointwise, a priori estimates for nonnegative solutions of anisotropic nonlinear elliptic equations. 1. Introduction In dimension n ≥ 2, given ??p = (p1, . . . , pn) with pi > 1 for i = 1, . . . , n, the anisotropic Laplace operator ∆??p is defined by ∆??p u = n∑ i=1 ∂ ∂xi ?pixiu , (1.1) where ?pixiu = |∂u/∂xi| pi?2 ∂u/∂xi. In this paper, we consider nonnegative solutions of anisotropic equations of the type ?∆??p u = f (u) (1.2) in open subsets of Rn, where f is continuous on R+. Anisotropic equations of type (1.2) have received much attention in recent years. They have been investigated by Alves–El Hamidi [1], Antontsev–Shmarev [3–7], Bendahmane–Karlsen [10–12], Bendahmane–Langlais–Saad [13], Cianchi [19], D'Ambrosio [21], Fragala–Gazzola–Kawohl [25], Fragala–Gazzola–Lieberman [26], El Hamidi–Rakotoson [22,23], El Hamidi–Vetois [24], Li [33], Lieberman [34,35], Miha˘ilescu– Pucci–Ra˘dulescu [38, 39], Miha˘ilescu–Ra˘dulescu–Tersian [40], and Vetois [51].

• anisotropic equations


• arbitrarily large maximum


• ∂u ∂xi ?


• isotropic critical


• ∂xi


• pi ?


• positive real


• pi?2 ∂u˜?


• exponent pcr

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• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

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A PRIORI ESTIMATES FOR SOLUTIONS OF ANISOTROPIC ELLIPTIC EQUATIONS

´ ˆ ´ JEROME VETOIS

Abstract.We prove universal, pointwise,a prioriestimates for nonnegative solutions of anisotropic nonlinear elliptic equations.

1.Introduction In dimensionn≥2, given − p → = (p 1 , . . . , p n ) withp i >1 fori= 1, . . . , n, the anisotropic Laplace operatorΔ −→ p is defined by n Δ − p u X ∂ x∂ i r x p i i u ,(1.1) → = i=1 wherer xp i i u=|∂u/∂x i | p i −2 ∂u/∂x i this paper, we consider nonnegative solutions of. In  anisotropic equations of the type −Δ −→ p u=f(u) (1.2) in open subsets ofR n , wherefis continuous onR + . Anisotropic equations of type (1.2) have received much attention in recent years. They have been investigated by Alves–El Hamidi [1], Antontsev–Shmarev [3–7], Bendahmane–Karlsen [10–12], Bendahmane–Langlais–Saad [13], Cianchi[19],D’Ambrosio[21],Fragal`a–Gazzola–Kawohl[25],Fragala`–Gazzola–Lieberman[26], ElHamidi–Rakotoson[22,23],ElHamidi–V´etois[24],Li[33],Lieberman[34,35],Mih˘ailescu– Pucci–Ra˘dulescu[38,39],Mih˘ailescu–Ra˘dulescu–Tersian[40],andV´etois[51].Theyhave strong physical background. Time evolution versions of these equations emerge, for instance, from the mathematical description of the dynamics of fluids in anisotropic media when the conductivities of the media are different in different directions. We refer to the extensive booksbyAntontsev–Dı´az–Shmarev[2]andBear[9]fordiscussionsinthisdirection.They also appear in biology as a model for the propagation of epidemic diseases in heterogeneous domains (see, for instance, Bendahmane–Karlsen [10] and Bendahmane–Langlais–Saad [13]). In connection with the anisotropic Laplace operator (1.1), for any open subsetΩinR n , we define the Sobolev space W 1 lo , c −→ p (Ω) =  u∈L 1 olc (Ω) ; u ∂x ∂ i ∈L l p o i c (Ω)∀i= 1, . . . , n  , where for any real numberp≥1,L l p oc (Ωfor the space of all measurable functions on) stands Ωwhich belong toL p (Ω 0 ) for all compact subsetsΩ 0 ofΩ references on anisotropic. Possible SobolevspacesareBesov[14],Hasˇkovec–Schmeiser[30],Kruzhkov–Kolod¯ıı˘[31],Kruzhkov– Korolev [32], Lu [37], Nikol 0 ineristhsndieWoc05.]si[idTro],an7,48ık[4´nsoka´R,]54[ı˘isk −→ paper weak solutions inW 1 lo , c p (Ω)∩L l ∞ oc (Ω As) of equations of type (1.2). a remark, we know by Lieberman [34, 35] that if the functionfis continuous, then any weak solution in 1 −→ 2) belongs toW 1 l ,∞ (Ω), and in particular, is continuous. W lo , c p (Ω)∩L l ∞ oc (Ω) of equation (1. oc Date:January 13, 2009.Revised:February 13, 2009. Published in Methods & Applications Theory,Nonlinear Analysis:71(2009), no. 9, 3881–3905. 1

A PRIORI ESTIMATES FOR SOLUTIONS OF ANISOTROPIC EQUATIONS

2

In this paper, we aim to find universal, pointwise,a prioriestimates for solutions of equations like (1.2). By universal, we mean that the estimate does not depend on the solution. In the classical case of the isotropic Laplace operator, it is well known since the work of Gidas– Spruck [28] that such estimates can be derived via rescaling arguments from a Liouville result. We state in Theorem 1.1 oura priori large part of the Aestimates in the anisotropic case. paper relies on establishing Liouville results associated with the nonlinear anisotropic equation (1.2). Theorem 1.2, see Section 4, is actually a Liouville result of the type of Mitidieri– Pohozˇaev[41–44],whereweprovenonexistenceforinequalities.Theorem1.3,seeSection5, is a Liouville result of the type of Gidas–Spruck [27] and Serrin–Zou [49], where we prove nonexistence for equations. We define the critical exponentp cr ( − p → to be the supremum of the real numbers) Qsuch that for anyqin (p + , Q), wherep + = max (p 1 , . . . , p n ), there does not exist any nontrivial, nonnegative solution inW 1 lo , c −→ p (R n )∩L lo ∞ c (R n ) of the equation −Δ −→ p u=λu q−1 ,(1.3) whereλis any positive real number, with the convention thatp cr ( −→ p) =p + in case such a real numberQ As a remark, does not exist.by an easy change of variable, we can takeλ= 1 in equation (1.3). In casep i = 2 fori= 1, . . . , n, namely in the case of the isotropic Laplace operator, by Gidas–Spruck [27], we getp cr = +∞in casen= 2 andp cr = 2n/(n−2) in case n≥ the anisotropic regime, our3. Ina prioriestimate states as follows. Theorem 1.1.Letn≥2, − p → = (p 1 , . . . , p n ), andqbe such that1< p i < q < p cr ( −→ p)for i= 1, . . . , n. Letλbe a positive real number andfbe a continuous function onR + satisfying f(u) =u q−1 (λ (1.4)+ o (1)) asu→+∞ there exist two positive constants. ThenΛ 1 =Λ 1 (n, − p → , f)andΛ 2 =Λ 2 (n, −→ p , f) such that for any open subsetΩofR n satisfyingΩ6=R n , any nonnegative solutionuin , W ocl 1 −→ p (Ω)∩L lo ∞ c (Ω)of equation(1.2)satisfies u(x)≤Λ 1 +Λ 2  y i ∈ n ∂ f i Ω = X n 1 |x i −y i | q p − i i p  −1 (1.5) for all pointsxinΩ. Moreover, we can takeΛ 1 = 0in casef(u) =λu q−1 . Whenp cr ( −→ p)> p + , Theorem 1.1 provides, in particular, universal,a prioribounds on com- pact subsets ofΩfor nonnegative weak solutions of equation (1.2). nontrivial solutions Such are proved to exist by Fragala–Gazzola–Kawohl [25] whenf(u) =λu q−1 . ` We are now led to the difficult question of estimating the critical exponentp cr ( − p → ). As already mentioned, this question was solved by Gidas–Spruck [27] in the case of the classical Laplace operator. We also refer to Serrin–Zou [49] for an extension of this result in the context of thep-Laplace operator. Theorems 1.2 and 1.3, we state our results concerning In the anisotropic case. In case P n i=1 p 1 i >1, we letp ∗ be the exponent defined by n n−1 −1 ,(1.6) p ∗ = P i=1 p 1 i andp ∗ be the anisotropic Sobolev critical exponent (see, for instance, Troisi [50]), namely p ∗ = P n i=1 n p 1 i 1 .(1.7) − We then get the following result.

A PRIORI ESTIMATES FOR SOLUTIONS OF ANISOTROPIC EQUATIONS 3 Theorem 1.2.Letn≥2and − p → = (p 1 , . . . , p n )be such thatp i >1fori= 1, . . . , n, and letp ∗ andp ∗ be as in(1.6)and(1.7). There holdp cr ( − p → ) = +∞in case P i n =1 p 1 i ≤1,p ∗ ≤p cr ( − p → ) in casep ∗ > p + and P i n =1 p 1 i >1, and finally,p cr ( −→ p)≤p ∗ in casep ∗ > p + and P i n =1 p 1 i >1. We can state another result as follows where we prove thatp cr ( − p → ) is a small perturbation of the isotropic critical exponentnp/(n−p) as − p→ → (p, . . . , p) with 2≤p≤(n+ 1)/2. A more general nonexistence result is given and commented in Section 4. Theorem 1.3.Letn≥3 any real number. Forpin[2,(n+ 1)/2], there holds np −→ p cr ( − p → ) n−p → asp i →pwithp i ≥2fori= 1, . . . , n, where − p= (p 1 , . . . , p n ). 2.Some comments

Under the notations in Theorem 1.1, increasing if necessary the constantΛ 2 , in the isotropic casep i =pfori= 1, . . . , n, we can rewrite estimate (1.5) as − u(x)≤Λ 1 +Λ 2 d(x, ∂Ω) q− p p ,(2.1) whered(x, ∂Ω) is the distance from the pointxto the boundary of the domainΩ case. In p= 2, namely in the case of the classical Laplace operator, some important references related to thea prioriimatest)1ra(e.2ua-tBedi,Gid[20]ncer],Da[n71e´ornoV–´Vre,]72[kcurpS–sa Pola´cˇik–Quittner–Souplet[46],andSerrin–Zou[49](thelasttworeferencesareconcerned with thep-Laplace operator, but in casep= 2, they both extend the results in [17, 20, 27] to more general nonlinearities). Our proof of Theorem 1.1 is inspired by the recent work of Pol´aˇcik–Quittner–Souplet[46]onthederivationofa prioriestimates from Liouville results. This technique is based on rescaling arguments together with a so-called doubling property. We observe that in the anisotropic case, the possible behaviors, allowed by our estimate (1.5), of the nonnegative weak solutions of equation (1.2) near a boundary depend on the geometry of this boundary, on its orientation, and not only on the distance to it as in (2.1). As an interesting particular case, Theorem 1.1 providesa prioriestimates near an isolated singularity. We point out that when no anisotropy is involved, namely whenp i =pfor i= 1, . . . , n, ifn > pandp ∗ < q < p ∗ , wherep ∗ =p(n−1)/(n−p) andp ∗ =np/(n−p), then an explicit nonnegative weak solution of equation (1.3) inR n \{0}withλ= 1 is given by 1−p q−p u(x) =C n,p
i= n X 1 |x i | p− p 1 ! , where −1 1 p p (q(n−p)−p(n−1))

q− p q−p C n,p = (q−p) p . q−p As is easily seen, in this case, the growth near the boundary in our estimate (1.5) is sharp. Whereas, these estimates are no more sharp in the case of the equation−Δu=u q−1 when 2< q≤2 ∗ . In this case, the local behavior near an isolated singularity was established by Lions [36] forqin (2,2 ∗ eeBidaut)(ssnoixeetehtntoron[-V´eoran15]fp-Laplace operator) and by Aviles [8] forq= 2 ∗ . Our last remark on Theorem 1.1 is that the nonexistence of nontrivial, nonnegative weak solution of equation (1.3) on the whole Euclidean space is a necessary condition. Indeed, if