7 0 0 2 Boundary singularities of positive solutions of some nonlinear n elliptic equations a J 3 2 Singularités au bord de solutions positives d’équations elliptiques ] non-linéaires P A . Marie-Françoise Bidaut-Véron, Augusto C. Ponce, Laurent Véron h t Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200 Tours, France a m [ 2 Abstract v 3 Westudythebehaviornearx0 ofanypositivesolutionof(E)−∆u=uq inΩwhichvanisheson∂Ω\{x0},where 2 Ω⊂RN is a smooth domain, q ≥(N +1)/(N −1) and x0 ∈∂Ω. Our results are based upon a priori estimates 1 of solutions of (E) and existence, non-existence and uniqueness results for solutions of some nonlinear elliptic 1 equations on the upper-half unit sphere. To cite this article: M.-F. Bidaut-Véron, A.C. Ponce, L. Véron, C. R. 1 Acad. Sci. Paris, Ser. I XXX (2006). 6 0 Résumé / h Nous étudions le comportement quand x tend vers x0 de toute solution positive de (E) −∆u = uq dans Ω qui at s’annule sur ∂Ω\{x0}, ou` Ω⊂RN est un domaine régulier, q≥(N +1)/(N −1) et x0 ∈∂Ω. Nos résultats sont m fondés surdesestimations apriori dessolutions de(E),et desrésultats d’existence,denon existenceet d’unicité desolutionsdecertaineséquationselliptiquesnonlinéairessurlademi-sphèreunité.Pour citercet article:M.-F. : v Bidaut-Véron, A.C. Ponce, L. Véron, C. R. Acad. Sci. Paris, Ser. I XXX (2006). i X r a Version française abrégéeSoit Ω un ouvert régulier de RN, N ≥4, tel que 0∈∂Ω. E´tant donnéq >1, nousconsidéronsunefonctionu∈C2(Ω)∩C(Ω\{0})quivérifie(3).Nousnousintéressonsàladescription du comportement de u au voisinage de 0. Email addresses: veronmf@univ-tours.fr (Marie-FrançoiseBidaut-Véron),ponce@lmpt.univ-tours.fr (AugustoC. Ponce), veronl@univ-tours.fr (LaurentVéron). PreprintsubmittedtoElsevierScience 2février2008 Nous distinguerons les trois valeurs critiques de q données par (4). Si 10. Si q ≤q 0. 1 2 Les démonstrations détaillées sont présentéesdans [2]. 1. Introduction and main results Let Ω be a smooth open subset of RN, N ≥ 4, such that 0 ∈ ∂Ω and let q > 1. Assume that u∈C2(Ω)∩C(Ω\{0}) is a solution of −∆u=uq in Ω, u≥0 in Ω, (3)   u=0 on ∂Ω\{0}. Our goal in this paper is to describe the behavior of u in a neighborhood of 0. This problem has similar features with the case where x ∈ Ω, which has been studied by Gidas- 0 Spruck [7]. In our case, we encounter three critical values of q in describing the local behavior of u: N +1 N +2 N +1 q := , q := and q := . (4) 1 2 3 N −1 N −2 N −3 When 10. If q ≤q 0 if 1 < q < q (see [1]), but R x0 1 k is necessarily equal to 0 if q ≥ q . Indeed, by the maximum principle, u satisfies u ≥kP (x,0), where 1 Ω P denotes the Poissonpotential of Ω. Since uq ∈L1(Ω) (by the Doob Theorem), we must have k =0 if Ω ρ q ≥q . 1 3 Detailed proofs will appear in [2]. 2. Sketch of the proofs Proof of Theorem 1.1. Assertion (i) is proved by multiplying (6) by φ(σ) = σ . Note that φ is the first 1 eigenfunction of −∆′ on SN−1, with eigenvalue λ = N −1. Integrating the resulting expression over + 1 SN−1, and using the fact that 11. Then, any solution of (7) satisfies N −3 (N −1)(q−1) N −1 (q−q ) |∇′v|2φdσ− λ+ v2φdσ = 3 q+1 ZS+N−1 q+1 (cid:18) q−1 (cid:19)ZS+N−1 ′ 2 =− |∇v| dτ. Z∂S+N−1 ′ ′ ′ This identity is obtained by computing the divergence of the vector field P = h∇φ,∇vi∇v, where ∇′ is the gradient on SN−1, and then using the fact that the first eigenfunction satisfies D2φ+φg =0, 0 where g is the tensor of the standard metric on SN−1. In order to establish (iii), it suffices to observe 0 that ℓ ≤−N−1 ⇐⇒q ≥q . N,q q−1 3 Proof of Theorem 1.2. We first notice that any positive solution of (7) depends only on the variable θ = cos−1(x /|x|) ∈ [0,π/2]; this follows from a straightforward adaptation of the Gidas-Ni-Nirenberg 1 moving plane method to SN−1 (see [9]). Thus, v satisfies + v′′+(N −2)cotθv′+λv+vq =0 in (0,π/2), (11)  ′ v (0)=0, v(π/2)=0.  Let w(θ):=sinαθv(θ), where α>0. By choosing α=2(N −2)/(q+3), then w satisfies  π/2 w′(π/2) 2 = G′(θ)w2(θ)dθ, (12) Z0 where G is a function of the form G(cid:0)(θ) = si(cid:1)nβ′θ α sin2θ+α ; the parameters α ,β′ ∈ R and α ≥ 0 1 2 1 2 can be explicitly computed in terms of λ, N and q. (cid:0) (cid:1) Assume, by contradiction, that v and v are two distinct solutions of (11). Then, 1 2 π/2 v v vq−1−vq−1 dθ =0. (13) 1 2 2 1 Z0 (cid:0) (cid:1) Therefore, their graphs must intersect at some θ ∈ (0,π/2). We claim that v and v intersect at least 0 1 2 twicein(0,π/2).Ifthereisonlyoneintersectionpoint,thenassumingv (0)>v (0)itcanbeshownthat 2 1 there existsγ ≥ w2′(π/2) 2 suchthatthe functionθ 7→G′(θ) w2(θ)−γw2(θ) is nonnegativein(0,π/2). w′(π/2) 2 1 1 Thus, by (12), (cid:16) (cid:17) (cid:0) (cid:1) π/2 0< G′(θ) w2(θ)−γw2(θ) dθ = w′(π/2) 2−γ w′(π/2) 2 ≤0. 2 1 2 1 Z0 (cid:2) (cid:3) (cid:0) (cid:1) (cid:0) (cid:1) 4 This is a contradiction. Therefore, v and v must intersect at least twice. This fact leads to another 1 2 contradiction by using the Shooting Method (see [8]). Thus, v =v in (0,π/2). 1 2 Remark 3 The method above follows the lines of the proof of Kwong-Li [8]. Proof of Theorem 1.3. It follows from methods developed in [7] and [3]. For simplicity, we shall assume that a=1 and ∂Ω∩B =∂RN ∩B . We set 1 + 1 w(t,σ)=r2/(q−1)u(r,σ), t=log(1/r)∈(0,∞)×SN−1 :=Q. + Then, w satisfies q+1 w − N −2 w +∆′w+ℓ w+wq =0 in Q (14) tt t N,q q−1 (cid:18) (cid:19) and w vanishes on (0,∞)×∂SN−1. Since w is uniformly bounded on Q, standard a priori estimates for + elliptic problems yield ∂k∇′jw ≤M in (1,∞)×SN−1 t k,j + for any integers k,j ≥ 0, where ∇′j stands for the covariant derivative on SN−1. Thus, the trajectory (cid:12) (cid:12) T = w(t,·) : t ≥ 1} is relatively(cid:12) compac(cid:12)t in C2 SN−1 . Multiplying (14) by w and integrating over w + t SN−1, we obtain + (cid:8) (cid:0) (cid:1) d q+1 H(t)= N −2 w2dσ, (15) dt (cid:18) q−1(cid:19)ZS+N−1 t where 1 2 H(t):= w2−|∇′v|2−ℓ w2+ wq+1 dσ. 2ZS+N−1(cid:18) t N,q q+1 (cid:19) Since q 6= q , we know that N − 2(q + 1)/(q − 1) 6= 0. Thus, iterated energy estimates imply that 2 w (t,·), w (t,·) → 0 in L2(SN−1) as t → ∞. Therefore, the limit set Γ of T is a connected subset of t tt + w the set of solutions of (6). By Theorem 1.1, we deduce that {0} if q =q or q ≥q , 1 3 Γ = w ({0} or {ω0} if q1 q . 1 We now consider the case q = q ; we borrow some ideas from [1] and [11]. We first prove, by ODE 1 techniques, that X(t):= w(t,·)φdσ ≤Ct−(N−1)/2. (16) ZS+N−1 Using (8) and the boundary Harnack inequality (see [5]), we derive 0≤w(t,σ)≤Ct−(N−1)/2 in (1,∞)×SN−1. (17) + Set η(t,σ):=t(N−1)/2w(t,σ). We verify as above that the limit set Γ in C2 SN−1 of the trajectory T η + η of η is an interval of the form κφ : 0 ≤ κ ≤ κ ≤ κ . In order to show that T is reduced to a single point, we prove that kr(t,·)k ≤Ct−1, wh0ere 1 (cid:0) η (cid:1) L2(cid:8) (cid:9) r(t,·):=η(t,·)−z(t)φ and z(t)= η(t,.)φdσ. ZS+N−1 Writing the equationsatisfiedby z as a non-homogeneoussecondorder linearODE, we provethat either z(t)→0, which implies that u is continuous at 0, or z(t)→k˜ as t →∞, for some constant depending N only on N. 5 Proof of Theorem 1.4. Itisanapplicationofthe DoublingLemmaMethodintroducedin[10],fromwhich we derive the following local estimate: Lemma 2.1 Let10 depending only on Ω. (cid:0) (cid:1) Apply this lemma with x ∈∂Ω\{0} and R=|x |/2. Using elliptic regularity theory, we obtain 0 0 0≤u(x)≤Cρ(x)|x|−2/(q−1)−1 ∀x∈Ω such that 0<ρ(x)<|x|/2. If ρ(x)≥|x|/2, then we use Gidas-Spruck’s internal estimates (see [7]). We thus obtain (10). Acknowledgements The authors are grateful to E. N. Dancer for his invaluable comments. References [1] M.-F.Bidaut-Véron,Th.Raoux, Asymptoticsofsolutionsofsomenonlinearellipticsystems,Comm.PartialDifferential Equations21(1986), 1035–1086. [2] M.-F.Bidaut-Véron,A.C.Ponce,L.Véron,inpreparation. [3] M.-F. Bidaut-Véron, L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emdenequations,Invent. Math.106(1991), 489–539. [4] M.-F. Bidaut-Véron, L. Vivier, An elliptic semilinear equation with source term involving boundary measures: the subcriticalcase,Rev.Mat.Iberoamericana16(2000), 477–513. [5] L.Cafarelli,E.Fabes,S.Mortola,S.Salsa, Boundarybehaviorofnonnegativesolutionsofellipticoperatorsindivergence form,IndianaUniv.Math.J.30(1981), 621–640. [6] J.Doob,Classicalpotential theoryanditsprobabilisticcounterpart, Springer,London,1984. [7] B.Gidas,J.Spruck, Globalandlocalbehaviorofpositivesolutionsofnonlinearellipticequations,Comm.PureAppl. Math.34(1981), 525–598. [8] M.K.Kwong,Y.Li, Uniquenessofradialsolutionsofsemilinearellipticequations,Trans.Amer.Math.Soc.333(1992), 339–363. [9] P.Padilla, Symmetrypropertiesofpositivesolutionsofellipticequationsonsymmetricdomains,Appl.Anal.64(1997), 153–169. [10] P. Pola´ˇcik, P. Quittner, Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville type theorems.PartI:Ellipticequations andsystems,DukeMath.J.,toappear. [11] L.Véron, Comportementasymptotiquedessolutionsd’équationselliptiquessemi-linéairesdansRN,Ann.Math.Pura Appl.127(1981), 25–50. 6

Boundary singularities of positive solutions of some nonlinear elliptic equations

Similar Boundary singularities of positive solutions of some nonlinear elliptic equations