CUBO A Mathematical Journal Vol.14, No¯ 02, (43–59). June 2012 A New proof of the CR Pohoz̆aev Identity and related Topics. Najoua Gamara Département de Mathématiques, Faculté des Sciences de Tunis, El Manar II 2092, Tunis, Tunisia. email: Najoua.Gamara@fst.rnu.tn Ali Ben Ahmed email: Ali.Ben.Ahmed@umpa.ens-lyon.fr and Aribi Amine email: Amine.Aribi@lmpt.univ-tours.fr ABSTRACT In this paper we give a new proof for “the CR Pohoz̆aev Identity” and deduce non existence results of positive solutions for semi-linear boundary value problems on star- shaped domains (P) { −∆Hu = g(u) in Ω u = 0 in ∂Ω, where ∆H is the sublaplacian of the Heisenberg group H n, g is a C1 function on a star-shaped and bounded domain Ω of Hn. RESUMEN En este art́ıculo presentamos una nueva demostración de la identidad de CR Pohozaev sobre el grupo de Heisenberg y deducimos resultados sobre la no existencia de solu- ciones positivas para problemas semi-lineales con valores en la frontera sobre dominios estrellados (P) { −∆Hu = g(u) in Ω u = 0 in ∂Ω, donde ∆H es el sublaplaciano del grupo de Heisenberg H n, g es una función de clase C1 sobre un dominio estrellado y acotado Ω de Hn. Keywords and Phrases: Analysis on CR manifolds, CR structure, CR Pohoz̆aev Identity, Crit- ical growth, Yamabe like problems. 2010 AMS Mathematics Subject Classification: 32V20; 32V05; 35H10; 35B20; 35J60; 22E30; 35B60; 35J65; 35B45. 44 Najoua Gamara, Ali Ben Ahmed and Aribi Amine CUBO 14, 2 (2012) 1 Introduction and Main Results We are concerned with non existence results for the following semilinear boundary value prob- lems on a bounded domain Ω of the Heisenberg group Hn (P) { −∆Hu = g(u) in Ω u = 0 in ∂Ω, where ∆H is the sublaplacian of H n, g is a C1 function. Recall that the Heisenberg group Hn is the homogeneous Lie group whose underlying manifold is R 2n+1 and group law given by τξ′(ξ) = ξ ′ · ξ = (x + x′, y + y′ t + t′ + 2(< x, y′ > − < x′, y >)) where < ., . > denotes the inner product in Rn, ξ = (x, y, t) and ξ′ = (x′, y′, t′). The homogeneous norm of the space Hn is ρ(ξ) = ( (| x |2 + | y |2)2 + t2 ) 1 4 and the natural distance is accordingly defined by d(ξ, ξ′) = ρ(ξ−1 · ξ′). The Koranyi ball of center ξ 0 and radius r for this distance is given by Br(ξ) = {ξ ∈ H n/ d(ξ0, ξ) < r}. There are a remarkable families of transformations groups on Hn, the group of parabolic dilations and the groups of left translations. The parabolic Hn-dilatations are the following transformations δλ : H n −→ Hn (x, y, t) −→ (λx, λy, λ2t) , λ > 0. The Jacobian determinant of δλ is λ 2n+2, it yields that the homogeneous dimension of Hn is Q = 2n + 2. For a given ξ ′ ∈ Hn, one can define a group of left translations by setting: τα(ξ) = ταξ′ (ξ) = αξ ′ · ξ, ∀ξ ∈ Hn The generators of the group of dilations {δλ, λ > 0} and the group of left translations {ταξ′ , α ∈ R} are given respectively by the following smooth vector fields: X = ∑ i=1 ( xi∂xi + yi∂yi ) + 2t∂t (1.1) CUBO 14, 2 (2012) A New proof of the CR Pohoz̆aev Identity 45 and Y(ξ′) = Y(x′, y′, t′) = ∑ i=1 ( x′i∂xi + y ′ i∂yi ) + (t′ + 2(< x, y′ > − < x′, y >))∂t. (1.2) We say that a function u is homogeneous of degree k with respect to the parabolic dilations {δλ, λ > 0} if and only if u ◦ δλ = λ ku for λ > 0, which implies that its Lie derivative with respect to X satisfies LXu = X u = k u. For example, the naturel distance function is homogenous of degree 1. In the other hand a function u is homogeneous of degree k with respect to the group of left translations {ταξ′ , α ∈ R} if and only if its Lie derivative with respect to Y satisfies LY(ξ′) u = Y(ξ ′) u = k u. The subelliptic gradient is given by ∇Hn = (X1, ..., Xn, Y1, ..., Yn) where Xi = ∂xi +2yi∂t, Yi = ∂yi −2xi∂t, i ∈ {1, 2...n} span the horizontal subspace of the tangent space of Hn accordingly to the following decomposition THn = H ⊕ RT, where H is the horizontal subspace and T is the Reeb vector field given by T = ∂t. The Lie Algebra of left invariant vector fields is generated by {(Xi, Yi)1≤i≤n, T}. Since [Xi, Yi] = −4T, the Heisenberg laplacian ∆H = n∑ i=1 (X2i + Y 2 i ), is a second order degenerate elliptic operator of Hörmander type and hence it is hypoelliptic. If we denote by A = (aij) the (2n + 1) × (2n + 1) symmetric matrix given by aij = δij if i, j = 1, ...2n, a(2n+1)j = −2xj if j = n + 1, ...., 2n, and a(2n+1)(2n+1) = 4|z| 2. We remark that the matrix A is related to ∆H by the formula ∆H = div(A ∇) where ∇ and div denote respectively the euclidian gradient and the euclidian divergence operator of R2n+1. The canonical contact and volume forms of Hn are given by θ0 = dt + 2 ∑ 1≤i≤n (xi dyi − yidxi) (1.3) 46 Najoua Gamara, Ali Ben Ahmed and Aribi Amine CUBO 14, 2 (2012) and dvθ0 = θ0 ∧ dθ n 0 . (1.4) A fundamental solution of −∆H with pole at zero is given by ( one can see [7]) Γ(ξ) = cQ d(ξ)Q−2 , where cQ = Γ2(n/2) 24−2nπn+1 and Q = 2n + 2. Moreover, a fundamental solution with pole at ξ is Γ(ξ, ξ ′ ) = cQ d(ξ, ξ ′ )Q−2 . A basic role in the functional analysis on the Heisenberg group is played by the following Sobolev- type inequality: |ϕ|2Q∗ ≤ c|∇Hnϕ| 2 2, ∀ϕ ∈ C ∞ 0 (H n) where Q∗ = 2Q Q−2 . This inequality ensures in particular that for every domain Ω of Hn, the function |ϕ| = |∇Hnϕ|2 is a norm on C∞0 (Ω). We denote by S 1,2(Ω) the closure of C∞0 (Ω) with respect to this norm, S1,2(Ω) becomes a Hilbert space with the inner product: < u, v >S1,2= ∫ Ω < ∇Hnu, ∇Hnv > dvθ0. Define S1,20 (Ω) as the completion of C ∞ 0 (Ω) with respect to the norm above. The Pohoz̆aev Identity is the principle tool used here to investigate the relation between do- main geometry and solvability of equation (P). We seek u a positive solution to equation (P), where g has critical or supercritical growth, meaning, g(u) ≥ ku1+ 2 n for some positive constant k. We ask the question ” for a prescribed domain and a nonlinearity g, can we find a positive solution u?”. For Euclidean domains Ω ⊂ RN, S.Pohoz̆aev in [19] proved that there is no solution for starlike ones, on the other hand, A.Bahri and J.M.Coron, W.Y.Ding in [1] and [6], have shown that a so- lution exists when g(u) = up∗, and the domain has nontrivial topology, here p∗ = (N + 2)/(N − 2) is the critical exponent for the compactness of the Sobolev inclusion W k,p 0 (Ω) →֒ Lq(Ω), for 1 q = 1 p − k n , 1 < p < q < ∞ where W k,p 0 (Ω) is the completion of C∞0 (Ω) with respect to the norm ‖u‖Wk,p(Ω) = Sup l(α)≤k‖D αu‖Lp(Ω). CUBO 14, 2 (2012) A New proof of the CR Pohoz̆aev Identity 47 For the Heisenberg group and using arguments related to the topology of the domain, G.Citti and F.Uguzzoni [5] following the work of A. Bahri and Coron, gave the Kohn Laplacian counter- part of the celebrated theorem in [1], and proved an existence result for Yamabe type problem on domains which have a nontrivial homology group (with Z2-coefficients), I.Birendili, I.Capuzzo Dolcetta and A.Cutri in [3] used blow up techniques to prove existence results, while in [22] F.Uguzzoni gave a non-existence result for equation (P) involving the critical exponent on halfs- paces of the Heisenberg group. We have also to mention the non existence results of E.Lanconelli and F.Uguzzoni on unbounded domains of the Heisenberg group in [14] and [15], and the existence of positives solutions on the Heisenberg group one can see [4] and[2]. For euclidian domains by strict-starlike, we mean that if x ∈ Rn and ν is the boundary normal, then on the boundary of the domain (x.ν) > 0 for all x. P.Pucci and J.Serrin noted that Pohoz̆aev’s result did not require strict starlikeness on the domain and what was needed was a domain with a vector function h that acted like the starlike vector field h = x. Several authors P.Pucci, J.Serrin, R.Schaaf, J.McGough, J.Mortesen, C.Rickett and G.Stubendieck in [20], [21], [16], [17] and [18] have examined this new class of h-starlike domains and the resulting extensions of the Pohoz̆aev like results. While for the Heisenberg group Hn using the geometry of the domain to give non existence and existence results for equation (P), N.Garofalo and E.Lanconelli in [11] have used the analogy with the hstarlike euclidean domains for a given vector field h. They defined for the Heisenberg group a notion of CR starlike domains for two special smooth vector fields, X and Y which are respectively the generator of the group of dilations and the generator of the group of left translations of Hn given by (1.1) and (1.2). Next we will introduce the definition given in [11] of domains starshape- ness which will be used throughout the present work. Given a piecewise C1 bounded domain Ω ⊂ Hn, we say that it is δ−starshaped with respect to a point ξ0 ∈ Ω, if denoting by N the outer unit normal to the boundary of τξ−1 0 (Ω), we have X.N ≥ 0 (1.5) at every point of ∂(τξ−1 0 (Ω)). For a bounded domain Ω of Hn, we denote by C(Ω) the space of all continuous functions f : Ω → R such that Xif, Yif, X 2 i f and Y 2 i f for i ∈ {1, 2, ...n} are continuous functions on Ω and continuous up to the boundary of Ω. Our main results are: - CR versions of the ”Pohoz̃aev identity”: 48 Najoua Gamara, Ali Ben Ahmed and Aribi Amine CUBO 14, 2 (2012) • Let u ∈ C(Ω) be a solution of the equation (P), then we have ∫ Ω |∇Hnu| 2X.Ndσ = −(Q − 2) ∫ Ω ug(u)du + 2Q ∫ Ω G(u)du. where G(u) = ∫u 0 g(s) ds. • We replace in equation (P) g(u) by g(ξ, u) = u1+ 2 n + h(ξ) u, with ξ ∈ Hn and h ∈ C∞(Hn), set (P ′) the equation thus obtained. If u ∈ C(Ω) is a solution of (P ′), then we have ∫ ∂Ω |∇Hnu| 2X.Ndσ = −2 ∫ Ω ( h + 1 2 (Xh) ) u2 dvθ0. - Pohoz̆aev’s non existence results: Let Ω ⊂ Hn be a bounded and connected domain such that 0 = (0, 0, 0) ∈ Ω and Ω is δ−starshaped with respect to this point. •Then any positive solution u of equation (P) vanishes identically if − (Q − 2)ug(u) + 2QG(u) ≤ 0. (1.6) • If g(u) = u1+ 2 n + λ u, λ ≤ 0, then (P) has no positive solution u different of the trivial solution u ≡ 0. • Let the function h given in equation (P ′) satisfies h + 1 2 (Xh) ≤ 0. (1.7) Then there is no positive solution u ∈ S1,20 (Ω) of equation (P ′) unless u ≡ 0. The paper is organized as follows. In section 2, we prove preliminary results and give the CR Pohoz̆aev Identity. The section 3 is devoted to establish some non existence result for equation (P) based on the theory of unique continuation property proved by N. Garofallo and E. Lanconelli for solutions of semi linear equations on Heisenberg group domains, one can see [10] and [11]. In section 4, we study a Yamabe like problem on a bounded domain of the Heisenberg group and deduce a non existence result using a related CR Pohoz̆aev Identity. 2 Description of the Problem We will be interested on the existence of a positive solution to the following semilinear equation CUBO 14, 2 (2012) A New proof of the CR Pohoz̆aev Identity 49 (P) { −∆Hu = g(u) in Ω u = 0 in ∂Ω, where ∆H is the sublaplacian of H n, g is a C1 function on Ω a bounded domain of the Heisenberg group Hn. Lemma 2.1. If u is a solution for problem (P), then we have − ∫ Ω ∆Hu(Xu) = ∫ Ω g(u)(Xu) = ∫ Ω X(G(u)) = −(2n + 2) ∫ Ω G(u) where G(u) = ∫u 0 g(s) ds. Proof: We multiply equation (P) by Xu and integrate by parts, we obtain − ∫ Ω ∆Hu(Xu) = ∫ Ω g(u)(Xu). Since ∂ ∂xi (xiG(u)) = G(u) + xi ∂ ∂xi G(u) for i ∈ {1, ...n}, we have ∫ Ω ∂ ∂xi (xiG(u)) = ∫ Ω G(u) + ∫ Ω xi ∂ ∂xi G(u), thus it yields that ∫ Ω G(u) + ∫ Ω xi ∂ ∂xi G(u) = 0, since u is equal to zero on the boundary of Ω. In the same way we obtain ∫ Ω G(u)+ ∫ Ω yi ∂ ∂yi G(u) = 0, for i ∈ {1, ...n} and ∫ Ω G(u)+ ∫ Ω t ∂ ∂t G(u) = 0, hence the proof of the lemma is complete. In what follows, for a bounded domain Ω of Hn, we denote by C(Ω) the space of all continuous functions f : Ω → R such that Xif, Yif, X 2 i f and Y 2 i f for i ∈ {1, 2, ...n} are continuous functions up to the boundary of Ω. Next we will consider the following vector field on Hn, P = Xu(∇Hn u) = (P1, P2, ...., P2n), where u is in C(Ω). If we denote by d̃iv the horizontal divergence operator on Hn, we remark that d̃ivP := divHnP = n∑ i=1 (XiP + YiP) = divP̃. (2.1) where P̃ = (P̃1, P̃2, ...., P̃2n, P̃2n+1) is the vector field on R 2n+1 obtained from P as P̃j = Pj, for j = 1, ...2n and P̃2n+1 = 2 n∑ j=1 (yjPj − xjPn+j). 50 Najoua Gamara, Ali Ben Ahmed and Aribi Amine CUBO 14, 2 (2012) Let Z be the vector field |∇Hnu| 2 X, since divX = 2n + 2, it yields ∫ Ω divZ = (2n + 2) ∫ Ω |∇Hnu| 2 + X < ∇u, A∇u > . (2.2) Using (8) and (9), we obtain the following result: Lemma 2.2. Let Ω be a bounded domain of Hn and u ∈ C(Ω). Then ∫ Ω d̃ivP = ∫ Ω Xu ∆Hu + ∫ Ω divZ − 2n ∫ Ω |∇Hnu| 2 − ∫ Ω < A∇u, ∇(Xu) > . Proof: We have d̃ivP = (Xu)d̃iv(∇Hnu) + ∇Hnu∇Hn(Xu) = Xu ∆Hu + ∇Hnu∇Hn(Xu). A simple computation gives P̃2n+1 = 2 n∑ j=1 (Xu) (yjXj − xjYj) therefore, since ∇Hnu∇Hn(Xu) =< ∇u, A∇Xu > and < ∇u, A∇Xu > = X < ∇u, A∇u > − < A∇u, n∑ j=1 ( X( ∂u ∂xi )∂xi + X( ∂u ∂yi )∂yi ) + X( ∂u ∂t )∂t) > + < ∇u, A∇u > −2 ∂u ∂t ( n∑ j=1 (yjXj(u) − xjYj(u) ) , we obtain ∫ Ω d̃ivP = ∫ Ω Xu ∆Hu + ∫ Ω divZ − (2n + 2) ∫ Ω |∇Hnu| 2 + ∫ Ω < A∇u, ∇u − n∑ j=1 ( X( ∂u ∂xi )∂xi + X( ∂u ∂yi )∂yi ) + X( ∂u ∂t )∂t) > − 2 ∫ Ω ∂u ∂t ( n∑ j=1 (yjXj(u) − xjYj(u)) = ∫ Ω Xu ∆Hu + ∫ Ω divZ − 2n ∫ Ω |∇Hnu| 2 − ∫ Ω < A∇u, ∇(Xu) > . Denoting by N the euclidian unit outer normal to ∂Ω and dσ the 2n-dimensional Hausdorff measure on R2n+1, if u is in C(Ω) the following holds CUBO 14, 2 (2012) A New proof of the CR Pohoz̆aev Identity 51 Theorem 2.1. 2 ∫ ∂Ω X(u)(A∇u.N)dσ − ∫ ∂Ω |∇Hnu| 2X.Ndσ = 2 ∫ Ω Xu∆Hu − 2n ∫ Ω |∇Hnu| 2. Proof: We have ∫ Ω divZdvθ0 = ∫ ∂Ω Z.Ndσ = ∫ ∂Ω < Z, N > dσ = ∫ ∂Ω |∇Hnu| 2(X.N)dσ, (2.3) and ∫ Ω d̃ivPdvθ0 = ∫ Ω divP̃dx = ∫ ∂Ω P̃.Ndσ, where P̃ = (P, 2 ∑ X(u)(yjXj(u) − xjYj(u))) = (Xu.∇Hn u, 2 n∑ i=1 (X(u)yjXj(u) − xjYj(u)) = X(u)(∇Hn u, 2 n∑ i=1 (yjXj(u) − xjYj(u)) = X(u)(A∇u). Therefore ∫ Ω divP̃dx = ∫ ∂Ω X(u)(A∇u.N)dσ. (2.4) On one hand, using Lemma 2.2 and (11), we obtain ∫ ∂Ω X(u)(A∇u.N)dσ = ∫ Ω Xu∆Hudvθ0 + ∫ ∂Ω |∇Hnu| 2X.Ndσ − 2n ∫ Ω |∇Hnu| 2dvθ0 − ∫ Ω < A∇u, ∇(Xu) > dvθ0. In the other hand, we have ∫ Ω d̃ivP = ∫ Ω divP̃ = ∫ Ω div(X(u)A∇u) = ∫ Ω (X(u)div(A∇u) + DX(u)(A∇u) = ∫ Ω (X(u)div(A∇u) + ∫ Ω ∇X(u).A∇u = ∫ Ω Xu.∆Hu + ∫ Ω < A∇u, ∇(Xu) > . The result follows. 52 Najoua Gamara, Ali Ben Ahmed and Aribi Amine CUBO 14, 2 (2012) We are now ready to state a CR version of the ”Pohoz̃aev identity”. Let g : R → R be a C1 function with primitive G(u) = ∫u 0 g(s)ds and let u ∈ C(Ω) be a solution of the equation (P) { −∆Hu = g(u) in Ω u = 0 in ∂Ω, in a bounded domain Ω ⊂ Hn. Then there hold ∫ Ω (−∆Hu)Xu = ∫ Ω g(u)X(u) = −(2n + 2) ∫ Ω G(u), and ∫ Ω |∇Hnu| 2 = ∫ Ω ug(u)du. (2.5) In the other hand X.u =< X, ∇u >, since the unit outer normal N = − ∇u |∇u| , we obtain X(u) = − < X, N > |∇u|. Therefore ∫ ∂Ω |∇Hnu| 2X.Ndσ = ∫ ∂Ω < A∇u, ∇u > .X.Ndσ = ∫ ∂Ω < A|∇u|N, |∇u|N > X.Ndσ and computing this product, one obtain < A∇u, ∇u >< X, N > = |∇u|2 < AN, N > . < X, N > = |∇u|2 < AN, N >< X, −∇u |∇u| > = −|∇u| < AN, N >< X, ∇u > = −|∇u| < AN, N > X.u = < A.∇u, N > X(u). It yields ∫ ∂Ω |∇Hnu| 2X.Ndσ = ∫ ∂Ω X(u)A∇u.Ndσ. (2.6) Therefore using (2.5) and (2.6), Theorem 2.3 reads as Theorem 2.2. Let u ∈ C(Ω) be a solution of the equation (P), then we have ∫ ∂Ω |∇Hnu| 2X.Ndσ = −(Q − 2) ∫ Ω ug(u)du + 2Q ∫ Ω G(u)du. CUBO 14, 2 (2012) A New proof of the CR Pohoz̆aev Identity 53 Theorem 2.4 is a CR version of the ”Pohoz̃aev identity”. 3 Pohoz̆aev’s non existence results We say that a family of functions has the unique continuation property, if no function besides possibly the zero function vanishes on a set of positive measure. In this section we proceed to establish some non existence result based on the theory of unique con- tinuation property proved by N. Garofallo and E. Lanconelli for solutions of semi linear equations on Heisenberg group domains, one can see [10] and [11]. We begin this section by introducing the notion of starshapeness which will be used throughout this paper. Definition 3.1. [11] Given a piecewise C1 domain Ω ⊂ Hn, we say that is δ−starshaped with respect to a point ξ0 ∈ Ω, if denoting by N the outer unit normal to the boundary of τξ−1 0 (Ω), we have X.N ≥ 0 (3.1) at every point of ∂(τξ−1 0 (Ω)). We observe that if we left-translate ξ0 to the origin then v(ξ) = u(τξ−1 0 ξ) is in Cτξ−1 0 (Ω) and satisfies the same equation as u. Therefore we may assume without loss of generality that the origin belongs to the domain Ω. By using the definition 3.1, we obtain as a consequence of theorem 2.4 the following non existence result for equation (P) Theorem 3.2. Let Ω ⊂ Hn be a connected and bounded domain containing 0 = (0, 0, 0), and assume that Ω is δ−starshaped with respect to this point. Then any positive solution u ∈ C(Ω) of equation (P) vanishes identically if − (Q − 2)ug(u) + 2QG(u) ≤ 0. (3.2) Proof: The proof is similar to the one given by N.Garofallo and E.Lanconelli for solution of such example of semi linear equations on Heisenberg group domains, one can see [11]. The proof is based on the theory of the unique continuation property developed in [10]. Since the domain is δ-starshaped i.e X.N ≥ 0 on the boundary of Ω, hence from theorem 2.4, we deduce that |∇Hnu| 2 is identically equal to 0 in ∂Ω ∩ Br(ξ̄) for some ξ̄ ∈ ∂Ω and r > 0. Therefore if we set u ≡ 0 in (Hn \ Ω̄) ∩ Br(ξ̄), we obtain a positive solution of − ∆Hu = Vu in Br(ξ̄) (3.3) 54 Najoua Gamara, Ali Ben Ahmed and Aribi Amine CUBO 14, 2 (2012) where ∆H is the sublaplacian of H n, V ∈ L∞(Br(ξ̄)), V = g(u) u when u 6= 0 and V = 0 when u = 0 in Br(ξ̄). In the appendix of [11] Corollary A.1, by using the method of the unique continuation property for the solution u of (16) the authors prove that u ≡ 0 in Br(ξ̄). We can reformulate the result of Corollary A.1 as follows, if we denote by D the maximal open set of Br(ξ̄) on which u vanishes then there exist a sphere S such its interior is entirely contained in D and there exist ξ ∈ ∂N ∩ S. As u vanishes in one side of S, it follows that ξ ∈ D, and hence the maximal open set D of Br(ξ̄) on which u vanishes is the hole ball i.e D = Br(ξ̄). To complete the proof i.e to show that u ≡ 0 on Ω, we use the fact that Ω is connected. Next we will focus on the special case where g(u) = λu + up ∗ , p∗ = 1 + 2 n is the critical exponent for the compactness of the Sobolev inclusion Sk,p(Ω) →֒ Ls(Ω), for 1 s = 1 p − k 2n + 2 , 1 < p < s < ∞; here Sk,p(Ω) is a Folland Stein space [12], the CR counterpart of The Sobolev space W1,2(Ω) for euclidean domains. Define S k,p 0 (Ω) as the completion of C∞0 (Ω) with respect to the norm ‖u‖Sk,p(Ω) = Sup l(α)≤k‖Z αu‖Lp(Ω), Z α = (Zα1, ......Zαk), where α = (α1, ......, αk), each αj is an integer 1 ≤ αj ≤ 2n, l(α) = α1 + ..... + αk and Zαj = { Xαj for 1 ≤ αj ≤ n Yαj for n + 1 ≤ αj ≤ 2n. More precisely, given λ ∈ R we would like to solve the problem Ep∗ (λ)    −∆Hu = u 1+ 2 n + λu in Ω u > 0 in Ω u = 0 in ∂Ω We obtain in this case the following non existence result Corollary 1. Suppose Ω is a bounded domain in Hn, which is δ−starshaped with respect to the origin 0 = (0, 0, 0) and let λ ≤ 0. Then any solution u ∈ S1,20 (Ω) of the boundary value problem Ep∗ (λ) vanishes identically. Proof: we will proceed by contradiction and suppose that there exist a nontrivial solution of Ep∗(λ). A simple computation shows that − (Q − 2)ug(u) + 2QG(u) = 2λ u2. (3.4) Therefore using the result of theorem 3.2, one deduce that λ > 0. The result follows. Let us remark that one can obtain the above result for a strict-δ−starshaped domain by a direct method, in fact two cases occur -If λ < 0, from equality (17) and theorem 2.4, we deduce that there is no positive solutions of CUBO 14, 2 (2012) A New proof of the CR Pohoz̆aev Identity 55 Ep∗ (λ). -If λ = 0, we use the Green formula for u, v ∈ C(Ω) ∫ Ω −∆Hu v dvθ0 = ∫ Ω ∇Hnu ∇Hnv dvθ0 − ∫ ∂Ω v A∇u.Ndσ (3.5) and set v ≡ 1 in (18), since N = −∇u |∇u| , we obtain for a solution u of (P) ∫ Ω −∆H u dvθ0 = ∫ ∂Ω |∇Hnu| 2 |∇u| dσ (3.6) Since Ω is strict-δ−starshaped with respect to 0 ∈ Hn, we have X.N(ξ) > 0 for all ξ ∈ ∂Ω. Thus from theorem 2.4, we deduce that |∇Hnu| 2 is identically equal to 0 on the boundary of Ω, therefore ∫ Ω −∆Hu = 0. (3.7) Hence ∫ Ω u1+ 2 n = 0, which means u = 0, since u ≥ 0. Remarks (1) The result of corollary 3.3 still hold true for supercritical value of the exponent p, ı.e p > p∗, for any value of λ < λ∗ = n(p − 1) − 2 p + 1 . (2) If the domain Ω is not δ−starshaped then equation (Ep) can have solutions even if (15) holds. In fact, if we choose a pseudo annulus Ω = {ξ = (x, y, t) ∈ Hn/R1 < x 2 + y2 < R2, |t| < T} for fixed R1, R2, T > 0, then for every fixed p > 1 and λ ≥ 0 the problem (Ep) has a positive solution u ∈ S1,20 (Ω) ∩ C ∞(Ω), which is Hölder continuous up to the boundary one can see [11]. However we can approch problem Ep∗(λ) by a direct method and attempt to obtain non-trivial solutions as relative minima of the functional Jλ(u) = 1 2 ∫ Ω (|∇Hnu| 2 − λ u2)θ0 ∧ dθ n 0 , (3.8) on the unit sphere of L2+ 2 n (Ω) ∑ = {u ∈ S1,20 (Ω), ‖u‖ 2+ 2 n L 2+ 2 n = 1}. (3.9) Equivalently, one may seek to minimize the Sobolev quotient Sλ(u) = ∫ Ω (|∇Hnu| 2 − λ u2)θ0 ∧ dθ n 0 ‖u‖ 2+ 2 n L 2+ 2 n , u 6= 0. (3.10) 56 Najoua Gamara, Ali Ben Ahmed and Aribi Amine CUBO 14, 2 (2012) Let us note that for λ = 0 S0(Ω) = inf u∈S1,2 0 (Ω), u 6=0 Sλ(u) = inf u∈S1,2 0 (Ω), u 6=0 ∫ Ω |∇Hnu| 2θ0 ∧ dθ n 0 ‖u‖ 2+ 2 n L 2+ 2 n , u 6= 0 (3.11) is related to the best constant for the Sobolev embedding S1,20 (Ω) →֒ L 2+ 2 n (Ω). 4 Yamabe like problems In the sequel we will consider the case where λ is a function. More precisely let h be a smooth function on Hn, we are looking for solutions of the semilinear equation on a bounded domain Ω Ep∗ (h)    −∆Hu = u 1+ 2 n + h u in Ω u > 0 in Ω u = 0 in ∂Ω This problem arises naturally in CR geometry, in fact let (M; θ) be a CR manifold of dimen- sion 2n + 1, n ≥ 1. We ask the question on whether there exist a contact form θ̃ on M conformal to θ i.e θ̃ = u 2 n θ, u > 0 which has a constant Webster scalar curvature. If we denote by Rθ (respectively R θ̃ ) the Webster scalar curvature of the contact form θ (respectively θ̃), we have the following relation: (2 + 2 n )∆b u + Rθu = Rθ̃ u 1+ 2 n (4.1) where ∆b is the sublaplacian ( the real part of the Kohn Spencer laplacian) of the manifold M. The existence of such a conformal contact form of constant Webster scalar curvature is equivalent to the existence of a positive solution of (4.1). This problem is known to be the Yamabe problem, one can see [12], [13], [8] and [9]. We have the following result. Lemma 4.1. If u is a solution of problem Ep∗(h), then ∫ Ω −∆Hu (Xu) dvθ0 = − ∫ Ω ( (n + 1) h + 1 2 X h ) u2 dvθ0 − n ∫ Ω u2+ 2 n dvθ0. 57 Proof: We multiply equation Ep∗ (h) by Xu and integrate by parts, we obtain ∫ Ω −∆Hu (Xu) = ∫ Ω h u(Xu) + ∫ Ω u1+ 2 n (Xu). on one hand, we have 2 (h u)(X u) = X(h u2) − (Xh) u2, (4.2) and a simple computation as done in Lemma 2.1 gives ∫ Ω X(h u2) = −(2n + 2) ∫ Ω h u2. (4.3) On the other hand, we have ∫ Ω u1+ 2 n (Xu) = −n ∫ Ω u2+ 2 n . (4.4) By using (26), (27) and (28), we obtain the desired result. Following the method used in section2, we obtain the CR version of the ”Pohoz̃aev identity” for the present case Lemma 4.2. Let u ∈ C(Ω) be a solution of the equation Ep∗ (h), then we have ∫ ∂Ω |∇Hnu| 2X.Ndσ = −2 ∫ Ω ( h + 1 2 (Xh) ) u2 dvθ0. Proof: Using theorem 2.3 and (13), we obtain ∫ Ω −∆Hu (Xu) = − 1 2 ∫ ∂Ω |∇Hnu| 2X.Ndσ − n ∫ Ω |∇Hnu| 2. (4.5) By comparing the result of lemma 4.1 and (29), the proof of lemma 4.2 is completed. We are now ready to state a non existence result for equation Ep∗(h). Corollary 2. Suppose Ω is a connected and bounded domain in Hn containing 0. Suppose that Ω is δ−starshaped with respect to this point and let h ∈ C∞(Hn) satisfying h + 1 2 (Xh) ≤ 0. (4.6) Then there is no positive solution u ∈ S1,20 (Ω) of equation Ep∗(h), u 6= 0. 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Introduction and Main Results Description of the Problem Pohoaev's non existence results Yamabe like problems Bibliography