No abstract

We address the question of attainability of the best constant in the following Hardy-Sobolev inequality on a smooth domain Ω of R n :when 0 < s < 2, 2 := 2 * (s) = 2(n−s) n−2 , and when 0 is on the boundary ∂Ω. This question is closely related to the geometry of ∂Ω, as we extend here the main result obtained in [GhK] by proving that at least in dimension n ≥ 4, the negativity of the mean curvature of ∂Ω at 0 is sufficient to ensure the attainability of µ s (Ω). Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions corresponding to the best constant in half-space, as well as a fine analysis of the asymptotic behaviour of appropriate minimizing sequences. The result holds true also in dimension 3 but the more involved proof will be dealt with in a forthcoming paper [GhR2].

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove thatThe technique is based on the existence of extremals of some Hardy-Sobolev type embeddings of independent interest. We also show that if u ∈ D p 1 (R n ) is a weak solution in R n of −∆pu − µ|x| −p |u| p−2 u = |x| −s |u| p ⋆ (s)−2 u + |u| q−2 u, then u ≡ 0 when either 1 < q < p ⋆ , or q > p ⋆ and u is also of class L ∞ loc (R n \ {0}).

Abstract. We are concerned in this paper with the bubbling phenomenon for nonlinear fourth-order four-dimensional PDE's. The operators in the equations are perturbations of the bi-Laplacian. The nonlinearity is of exponential growth. Such equations arise naturally in statistical physics and geometry. As a consequence of our theorem we get a priori bounds for solutions of our equations.We are concerned in this paper with understanding the bubbling phenomenon for fourth-order four-dimensional PDE's of exponential growth. Such equations arise naturally in statistical physics and in geometry (see [7] and [9]). In what follows, we let (M, g) be a smooth compact Riemannian 4-manifold without boundary. We also let (b ε ) ε>0 and (f ε ) ε>0 be sequences of smooth functions on M , and we let (A ε ) ε>0 be a sequence of smooth (2, 0)-symetric tensor fields. We assume that (b ε ), (f ε ) and (A ε ) converge as ε → 0 in the C k -topologies, k a positive integer, to limiting objects of the same nature, b 0 , f 0 and A 0 . Then we consider sequences (u ε ) ε>0 of solutions ofwhere) is the Laplace-Beltrami operator andFollowing standard terminology, we say that the u ε 's blow up if u ε (x ε ) → +∞ as ε → 0 for a sequence (x ε ) of points in M . We letbe the limit operator in (1). At last, we let G be the Green function of L 0 . The Green function is unique up to a constant when the kernel of L 0 consists only of constants. We write G as

Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -equivalently -a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems.It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n , the negative part of the corresponding Green's function is "small" when compared with its singular positive part, provided n ≥ 3.Moreover, the biharmonic Green's function in balls B ⊂ R n under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3.

The Paneitz operator discovered in [11] is the fourth order operator defined on a 4-dimensional Riemannian manifold (M, g) bywhere ∆ g u = −div g ∇u is the Laplacian of u with respect to g, S g is the scalar curvature of g, and Rc g is the Ricci curvature of g. An extension to manifolds of dimension n ≥ 5, due to Branson [2], is the fourth order operator defined byBoth P 4 g and P n g have conformal properties: for all u ∈ C ∞ (M ), P 4 g u = e −4ϕ P 4 g u when n = 4 andg = e 2ϕ g, while P n g (uϕ) = ϕ (n+4)/(n−4) P ñ g u when n ≥ 5 and g = ϕ 4/(n−4) g. With respect to these relations, P 4 g in dimension 4 is a natural analogue of ∆ g in dimension 2, while P n g in dimension n ≥ 5 is a natural analogue of the conformal Laplacian ∆ g + n−2 4(n−1) S g in dimension n ≥ 3. Possible references on the subject are the survey articles [3] by Chang, and [4] by Chang and Yang.We let here (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 5, and say that a fourth order operator P g is a Paneitz type operator with constant coefficients if P g u = ∆ 2 g u + α∆ g u + au (0.1)

We establish -among other things-existence and multiplicity of solutions for the Dirichlet problem i ∂ ii u+ |u| 2 ⋆ −2 u |x| s = 0 on smooth bounded domains Ω of R n (n ≥ 3) involving the critical Hardy-Sobolev exponent 2 ⋆ = 2(n−s) n−2 where 0 < s < 2, and in the case where zero (the point of singularity) is on the boundary ∂Ω. Just as in the Yamabe-type non-singular framework (i.e., when s = 0), there is no nontrivial solution under global convexity assumption (e.g., when Ω is star-shaped around 0). However, in contrast to the nonsatisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of ∂Ω at 0 in at least one direction. More precisely, we need the principal curvatures of ∂Ω at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of ∂Ω at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case.1 1 In our context, we specify the orientation of ∂Ω in such a way that the normal vectors of ∂Ω are pointing outward from the domain Ω.

We investigate the asymptotic behavior as k → +∞ of sequences (u k ) k∈N ∈ C 4 (Ω) of solutions of the equations ∆ 2 u k = V k e 4u k on Ω, where Ω is a bounded domain of R 4 and lim k→+∞ V k = 1 in C 0 loc (Ω). The corresponding 2-dimensional problem was studied by Brézis-Merle and Li-Shafrir who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Struwe and the author [1], such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on ∆u k , we recover such a quantization as in dimension 2.