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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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 andMps(R) be the vector space of all symmetricp×preal matrices. Namely, Mps(R) is the vector space ofp×preal matricesS= (Sij) which are such thatSij=Sjifor alli, j. ForA:M→Mps(R) smooth,A= (Aij), we consider vector valued equations like ΔpgU+A(x)U=12∗DU|U |2∗,(1.1) whereU:M→Rpis a map, referred to as ap-mapin order to underline the fact that the target space isRp,Δpgis the Laplace–Beltrami operator acting onp-maps, 2∗= 2n/(n−2), andDUis the derivation operator with respect toU. WritingU= (u1, . . . , up), we get |U |2∗=Ppi=1|ui|2∗,21∗DU|U |2∗=|ui|2∗−2uii, andΔgpU= (Δgui)i, whereΔg=−divgris 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 Δgui+XAij(x)uj=|ui|2∗−2ui,(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, ifH12is the Sobolev space of functions inL2with one derivative inL2, then 2∗ is the critical Sobolev exponent for the embeddings ofH12 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 spaceH12,p(M) of all p-maps whose components belong toH12(M), and we say that ap-mapUinH12,p(M) is a solution of (1.1)–(1.2) if its componentsuisolve (1.2) weakly fori= 1, . . . , n regularity. By theory, see Hebey [27], the components of any weak solution belong toC2,θ(M) for all real numbersθin (0, define the energy of a solution1). WeUof (1.1)–(1.2) by E(U) =i=pX Z|ui|2∗dvg,(1.3) 1M
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 theui’s are the components ofU, anddvgis the volume element of the manifold (M, g). We also define the functionalIA,gacting onH12,p(M) by 1 IA,g(U) = 2ZMiXp1|rui|2gdvg+12ZM,ijXp=1Aijuiujdvg = −12∗piX ZM|ui|2∗dvg.(1.4) =1 Critical points ofIA,g Weare solutions of the system (1.1)–(1.2). set
µA,g=Uin∈fNIA,g(U),(1.5) whereNis the Nehari manifold of the functionalIA,gdefined as the set ofp-mapsUin H12,p(M)\ {0}such thatDIA,g(U).U say that the operator We= 0.Δpg+Aiscoercive onH12,p(M) if its energy controls theH12,p-norm. A precise definition is given in Section 2. WhenΔpg+Ais coercive onH12,p(M), the lower boundµA,gis positive. Following standard terminology we say that a mapA:M→Mps(R) iscooperativeif its off-diagonal components are nonnegative. In other words,Asaid to be cooperative if there holdsis Aij≥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{i1, . . . , ik} and{j1, . . . , jk0},k+k0=p, such that there holdsAiαjβ≡0 inMfor allα= 1, . . . , kand β= 1, . . . , k0. 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 letKnbe the sharp constant for the embedding ofH˙12(Rn) intoL2∗(Rn). Then, as is well known, Kn=sn(n−4)2ω2n/n,(1.6) whereωnis the volume of the unitn-sphere. WhenΔgp+Ais coercive onH12,p(M), thep-map U) +21∗E( WU=2IA,g(E(U)U) !(n−2)/4U(1.7) belongs toNfor allU ∈H12,p(M)\{0}, whereIA,gis as in (1.4), andEis the energy function as in (1.3). In particular, for anyU ∈H12,p(M)\{0}, we get by (1.5) thatµA,g≤IA,g(WU), whereWUis as in (1.7), and it follows that 1 µA,g≤nUin∈fH2IA,gE(U()U)+2/212∗∗E(U) !n/2≤1nKn−n(1.8) for all (M, g) and allAsuch thatΔgp+Ais coercive, whereH=H12,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 ofH12(Rn) intoL2∗(Rn). The first result we prove is as follows.
