Hichem Chtioui

Existence results for the prescribed Scalar curvature on S^{3}@@@Résultats d’existence pour la courbure scalaire prescrite sur S^{3}

— This paper is devoted to the existence of conformal metrics on S3 with prescribed scalar curvature. We extend well known existence criteria due to Bahri-Coron. Résumé. — Ce papier est consacré à l’existence des métriques conforme sur S3 avec courbure scalaire prescrite. Nous étendons les critères d’existence bien connus de Bahri-Coron.

On the Prescribed Paneitz Curvature Problem on the Standard Spheres

Abstract In this paper, we study some fourth order conformal invariants on the standard n-spheres, n ≥ 5. Using topological arguments, we give a variety of classes of functions that can be realized as Paneitz curvature.

PRESCRIBING MEAN CURVATURE ON 𝔹n

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹n, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

Multiplicity and Existence Results for a Nonlinear Elliptic Equation With Sobolev Exponent

Abstract In this paper we consider the following nonlinear elliptic equation with Dirichlet boundary conditions: -Δu = K(x)up, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth domain in ℝn, n ≥ 4 and is the critical Sobolev exponent. Using dynamical and topological methods involving the study of critical points at infinity we establish, under generic conditions on K, some existence and multiplicity results.

Topological Methods for Boundary Mean Curvature Problem on Bn

Abstract Using an algebraic topological method and the tools of the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the the n-dimensional balls.

On a geometric equation involving the Sobolev trace critical exponent

In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of Rn, n≥4. Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness, and we provide a new existence result of Bahri-Coron type. Moreover, we establish, under generic boundary condition, a Morse inequality at infinity, which gives a lower bound on the number of solutions to the above problem.MSC:58E05, 35J65, 53C21, 35B40.

Existence of Conformal Metrics with Prescribed Q-Curvature

We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere . Under the assumption that the order of flatness at critical points of prescribed Q-curvature function is , we give precise estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.

ON A BI-HARMONIC EQUATION INVOLVING CRITICAL EXPONENT: EXISTENCE AND MULTIPLICITY RESULTS ∗

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ2u = K(x)up, u > 0 in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth domain in ℝn,n≥5, and p+1=2nn−4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.

On a fractional Nirenberg type problem on the n-dimensional sphere

In this paper, we study a fractional Nirenberg type problem involving the fractional Laplacian on the standard n-dimensional sphere, . We describe the lack of compactness of the associated variational problem and we give an existence result generalizing the one given by T. Jin, Y. Li and J. Xiong. Our method hinges on a readapted characterization of critical points at infinity techniques introduced by A. Bahri and Bahri–Coron.

Topological tools for the prescribed scalar curvature problem on S n

In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere Sn, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].