MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC SYSTEMS IN POTENTIAL FORM

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MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC SYSTEMS IN POTENTIAL FORM EMMANUEL HEBEY AND JEROME VETOIS Abstract. We discuss and prove existence of multiple solutions for critical elliptic systems in potential form on compact Riemannian manifolds. 1. Introduction Let (M, g) be a smooth, compact Riemannian n-manifold, n ≥ 3. Let also p ≥ 1 be a natural number and M sp (R) be the vector space of all symmetric p? p real matrices. Namely, M sp (R) is the vector space of p ? p real matrices S = (Sij) which are such that Sij = Sji for all i, j. For A : M ?M sp (R) smooth, A = (Aij), we consider vector valued equations like ∆pgU + A(x)U = 1 2? DU |U| 2? , (1.1) where U : M ? Rp is a map, referred to as a p-map in order to underline the fact that the target space is Rp, ∆pg is the Laplace–Beltrami operator acting on p-maps, 2 ? = 2n/(n ? 2), and DU is the derivation operator with respect to U . Writing U = (u1, . . . , up), we get |U|2 ? = ∑p i=1 |ui| 2? , 12?DU |U| 2? = ( |ui| 2??2 ui ) i , and ∆pgU = (∆gui)i, where

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Sujets

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• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

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MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC SYSTEMS IN POTENTIAL FORM ´ ˆ ´ EMMANUEL HEBEY AND JEROME VETOIS

Abstract.We discuss and prove existence of multiple solutions for critical elliptic systems in potential form on compact Riemannian manifolds.

1.Introduction

Let (M, g) be a smooth, compact Riemanniann-manifold,n≥3. Let alsop≥1 be a natural number andM ps (R) be the vector space of all symmetricp×preal matrices. Namely, M ps (R) is the vector space ofp×preal matricesS= (S ij ) which are such thatS ij =S ji for alli, j. ForA:M→M ps (R) smooth,A= (A ij ), we consider vector valued equations like Δ p g U+A(x)U= 1 2 ∗ D U |U | 2 ∗ ,(1.1) whereU:M→R p is a map, referred to as ap-mapin order to underline the fact that the target space isR p ,Δ p g is the Laplace–Beltrami operator acting onp-maps, 2 ∗ = 2n/(n−2), andD U is the derivation operator with respect toU. WritingU= (u 1 , . . . , u p ), we get |U | 2 ∗ = P p i=1 |u i | 2 ∗ , 2 1 ∗ D U |U | 2 ∗ =  |u i | 2 ∗ −2 u i  i , andΔ g p U= (Δ g u i ) i , whereΔ g =−div g ris the Laplace–Beltrami operator for functions. Another way in which we can write (1.1) is like in the form of the following elliptic system p Δ g u i + X A ij (x)u j =|u i | 2 ∗ −2 u i ,(1.2) j=1

where the equations have to be satisfied inM, and for alli= 1, . . . , p say that the. We system is of orderp, and refer to it as ap-system in potential form because of the nature of the nonlinearity. The system has a variational structure. It is also critical from the Sobolev viewpoint since, ifH 12 is the Sobolev space of functions inL 2 with one derivative inL 2 , then 2 ∗ is the critical Sobolev exponent for the embeddings ofH 12  Ininto Lebesgue spaces. casep= 1, (1.1)–(1.2) reduces to Yamabe-type equations, and we regard (1.1)–(1.2) as a natural extension of such equations to weakly coupled systems. We introduce the Sobolev spaceH 12 ,p (M) of all p-maps whose components belong toH 12 (M), and we say that ap-mapUinH 12 ,p (M) is a solution of (1.1)–(1.2) if its componentsu i solve (1.2) weakly fori= 1, . . . , n regularity. By theory, see Hebey [27], the components of any weak solution belong toC 2,θ (M) for all real numbersθin (0, define the energy of a solution1). WeUof (1.1)–(1.2) by E(U) = i= p X Z |u i | 2 ∗ dv g ,(1.3) 1 M

Date: January 19, 2007. Published inCommunications on Pure and Applied Analysis7(2008), no. 3, 715–741. 1

MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC SYSTEMS

2

where theu i ’s are the components ofU, anddv g is the volume element of the manifold (M, g). We also define the functionalI A,g acting onH 12 ,p (M) by 1 I A,g (U) =  2 Z M i X p 1 |ru i | 2 g dv g + 1 2 Z M ,ij X p =1 A ij u i u j dv g = − 1 2 ∗ p i X Z M |u i | 2 ∗ dv g .(1.4) =1 Critical points ofI A,g  Weare solutions of the system (1.1)–(1.2). set

µ A,g = U in ∈ f N I A,g (U),(1.5) whereNis the Nehari manifold of the functionalI A,g defined as the set ofp-mapsUin H 12 ,p (M)\ {0}such thatDI A,g (U).U say that the operator We= 0.Δ p g +Aiscoercive onH 12 ,p (M) if its energy controls theH 12 ,p -norm. A precise definition is given in Section 2. WhenΔ p g +Ais coercive onH 12 ,p (M), the lower boundµ A,g is positive. Following standard terminology we say that a mapA:M→M ps (R) iscooperativeif its off-diagonal components are nonnegative. In other words,Asaid to be cooperative if there holdsis A ij ≥0 inM for all distinct indicesiandj . Stillfollowing standard terminology, we say that (1.1)–(1.2) isfully coupledif the index set{1, . . . , p}does not split into two disjoint subsets{i 1 , . . . , i k } and{j 1 , . . . , j k 0 },k+k 0 =p, such that there holdsA i α j β ≡0 inMfor allα= 1, . . . , kand β= 1, . . . , k 0 . When (1.1)–(1.2) is not fully coupled, permuting if necessary the equations,A may be written in diagonal blocks and thep-system may split into two independent systems. Ap-map is said to bepositive what follows we associateif its components are all positive. In each solution of equation (1.1)–(1.2) with its opposite one, and call that a pair of solutions. A pair (U,−U) is said to be positive if eitherUor−Uis positive. We letK n be the sharp constant for the embedding ofH ˙ 12 (R n ) intoL 2 ∗ (R n ). Then, as is well known, K n = s n(n− 4 )2ω 2n/n ,(1.6) whereω n is the volume of the unitn-sphere. WhenΔ g p +Ais coercive onH 12 ,p (M), thep-map U) + 2 1 ∗ E( W U =
2  I A,g ( E(U) U)  ! (n−2)/4 U(1.7) belongs toNfor allU ∈H 12 ,p (M)\{0}, whereI A,g is as in (1.4), andEis the energy function as in (1.3). In particular, for anyU ∈H 12 ,p (M)\{0}, we get by (1.5) thatµ A,g ≤I A,g (W U ), whereW U is as in (1.7), and it follows that 1 µ A,g ≤ n U in ∈ f H
2  I A,g E (U ( ) U) + 2/2 1 2 ∗ ∗ E(U)  ! n/2 ≤ 1 n K n −n (1.8) for all (M, g) and allAsuch thatΔ g p +Ais coercive, whereH=H 12 ,p (M)\{0} second. The inequality in (1.8) follows from standard developments on the Yamabe problem, as in Aubin [4], by testingpwhich we choose to be like minimizers-maps with components all zero, except one ˙ for the embedding ofH 12 (R n ) intoL 2 ∗ (R n ). The first result we prove is as follows.