Wael Abdelhedi

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.

A complete study of the lack of compactness and existence results of a fractional Nirenberg equation via a flatness hypothesis, I

In this paper, we consider a nonlinear critical problem involving the fractional Laplacian operator arising in conformal geometry, namely the prescribed �-curvature problem on the standard n- sphere n ≥ 2. Under the assumption that the prescribed function is flat near its critical points, we give precise estimates on the losses of the compactness and we provide existence results. In this first part, we will focus on the case 1 < � ≤ n − 2�, which was not included in the results of Jin, Li and Xiong (14) and (15). MSC 2000: 35J60, 35B33, 35B99, 35R11, 58E30.

Concentration of solutions for the mean curvature problem

Abstract We consider the problem of conformal metrics equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball n , n ≥ 4. By variational methods, we prove the existence of two peak solutions that concentrate around a strict local maximum points of the mean curvature under certain conditions.

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.