arXiv subjects

El Hadji Abdoulaye Thiam

Publications and source records attributed to El Hadji Abdoulaye Thiam.

12 recordsLinked to original sources

arXiv2026

A Doubly Critical Elliptic Problem with Submanifold Singularities

Let $N \ge 4$, $Ω$ be a bounded domain in $\mathbb{R}^N$, and let $Σ\subset Ω$ be a smooth closed submanifold of dimension $k$ with $2 \le k \le N-2$. We study the existence of positive solutions $u \in H_0^1(Ω)$ to the Euler--Lagrange equation \[ -Δu + h u = λ\, ρ_Σ^{-s_1}\, u^{2^{*}_{s_1}-1} + ρ_Σ^{-s_2}\, u^{2^{*}_{s_2}-1} \quad \text{in } Ω, \] where $h : Ω\to \mathbb{R}$ is a continuous potential, $λ> 0$ is a real parameter, and $0 \le s_2 < s_1 < 2$. For $i=1,2$, the exponents \[ 2^{*}_{s_i} = \frac{2(N - s_i)}{N - 2} \] correspond to Hardy--Sobolev critical growth, and $ρ_Σ = \mathrm{dist}(\,\cdot\,, Σ)$ denotes the distance to the submanifold $Σ$. The problem involves two Hardy-type singular nonlinearities with different critical exponents, leading to a lack of compactness. Using variational methods, in particular the mountain pass lemma, together with a suitable construction of test functions, we prove existence results under appropriate assumptions. Our analysis shows that the local geometry of $Σ$ and the behavior of the potential $h$ near $Σ$ play a crucial role in the existence of positive solutions for this doubly critical problem.

math.AP↗

arXiv2025

A Nonlinear elliptic PDE with curve singularity on the boundary

Let $Ω$ be a bounded domain of $\mathbb{R}^{N+1}$ ($N \geq 3$) with smooth boundary $\partial Ω$ and $Σ$ be a closed submanifold contained on $\partial Ω$ and containing $0$. We are interesting in the existence of positive $H^1(Ω)$-solution of the following Hardy-Sobolev trace type equation \begin{equation*} \begin{cases} -Δu+u=0 \qquad & \textrm{ in $Ω$}\\\\ \displaystyle\frac{\partial u}{\partial ν}= ρ_Σ^{-s} u^{q_s-1} \qquad & \textrm{ on $\partial Ω$}, \end{cases} \end{equation*} where $ν$ is the unit outer normal of $\partial Ω$, $ρ_Σ: \partial Ω\to \mathbb{R}$ is the distance function in $\partial Ω$ to the curve $Σ$: $$ ρ_Σ(x):= \inf_{y \in Σ} d_{\tilde{g}}(x, y) $$ and for $0\leq s <1$, $q_s:=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev exponent. The existence of solution may depend on the local geometry of the boundary $\partial Ω$ and $Σ$ at $0$ or in the shapes of the domain $Ω$ and its boundary $\partial Ω$.

math.AP↗

arXiv2024

Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents

For $N\geq 4$, we let $Ω$ be a bounded domain of $\mathbb{R}^N$ and $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the equation \begin{equation}\label{Atusi} -Δu+hu=λρ^{-s_1}_Γu^{2^*_{s_1}-1}+ρ^{-s_2}_Γu^{2^*_{s_2}-1} \qquad \textrm{ in } Ω\end{equation} where $h$ is a continuous function and $ρ_Γ$ is the distance function to $Γ$. We prove the existence of a mountain pass solution for this Euler-Lagrange equation depending on the local geometry of the curve and the potential $h$.

math.AP↗

arXiv2023

Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents

We let $Ω$ be a bounded domain of $\mathbb{R}^3$ and $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the equation $$ -Δu+hu=λρ^{-s_1}_Γu^{5-2s_1}+ρ^{-s_2}_Γu^{5-2s_2} \qquad \textrm{ in } Ω$$ where $h$ is a continuous function and $ρ_Γ$ is the distance function to $Γ$. We prove existence of solutions depending on the regular part of the Green function of linear operator. We prove the existence of positive mountain pass solutions for this Euler-Lagrange equation depending on the mass which is the regular part of the Green function of the linear operator $-Δ+h$.

math.AP↗

arXiv2021

Influence of an $L^p$-perturbation on Hardy-Sobolev inequality with singularity a curve

We consider a bounded domain $Ω$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$ continuous functions on $Ω$. Let $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the perturbed Hardy-Sobolev equation: $$ -Δu+h u+bu^{1+δ}=ρ^{-σ}_Γu^{2^*_σ-1} \qquad \textrm{ in } Ω, $$ where $2^*_σ:=\frac{2(N-σ)}{N-2}$ is the critical Hardy-Sobolev exponent, $σ\in [0,2)$, $0< δ<\frac{4}{N-2}$ and $ρ_Γ$ is the distance function to $Γ$. We show that the existence of minimizers does not depend on the local geometry of $Γ$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-Δ+h$ and or on $b$. This is due to the perturbative term of order ${1+δ}$.

math.AP↗

arXiv2021

Nonlocal diffusion of smooth sets

We consider normal velocity of smooth sets evolving by the $s-$fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for $s\in [\frac{1}{2}, 1)$ while, for $s\in (0, \frac{1}{2})$, it is nearly proportional to the fractional mean curvature of the initial set. Our results show that the motion by (fractional) mean curvature flow can be approximated by fractional heat diffusion and by a diffusion by means of harmonic extension of smooth sets.

math.AP↗

arXiv2020

The role of the mean curvature in a mixed Hardy-Sobolev trace inequality

Let $Ω$ be a smooth bounded domain of $\mathbb{R}^{N+1}$ of boundary $\partial Ω= Γ_1 \cup Γ_2$ and such that $\partial Ω\cap Γ_2$ is a neighborhood of $0$, $h \in \mathcal{C}^0(\partial Ω\cap Γ_2) $ and $s \in [0,1)$. We propose to study existence of positive solutions to the following Hardy-Sobolev trace problem with mixed boundaries conditions \begin{align} \begin{cases} Δu= 0& \qquad \textrm{ in } Ω\\\ u=0 & \qquad \textrm{ on } Γ_1 \\\ \frac{\partial u}{\partial ν}=h(x) u + \frac{u^{q(s)-1}}{d(x)^{s}} & \qquad \textrm{ on } Γ_2, \end{cases} \end{align} where $q(s):=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev trace exponent and $ν$ is the outer unit normal of $\partial Ω$. In particular, we prove existence of minimizers when $N \geq 3$ and the mean curvature is sufficiently below the potential $h$ at $0$.

math.AP↗

arXiv2018

Multiply-periodic hypersurfaces with constant nonlocal mean curvature

We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of smooth branches of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.

math.AP↗

arXiv2018

Hardy-Sobolev inequality with higher dimensional singularity

For $N\geq 4$, we let $Ω$ to be a smooth bounded domain of $\mathbb{R}^N$, $Γ$ a smooth closed submanifold of $Ω$ of dimension $k$ with $1\leq k \leq N-2$ and $h$ a continuous function defined on $Ω$. We denote by $ρ_Γ\left(\cdot\right):=\dist_g\left(\cdot, Γ\right)$ the distance function to $Γ$. For $σ\in (0,2)$, we study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the nonlinear equation $$ -Δu+h u=ρ_Γ^{-σ} u^{2^*(σ)-1} \qquad \textrm{in } Ω, $$ where $2^*(σ):=\frac{2(N-σ)}{N-2}$ is the critical Hardy-Sobolev exponent. In particular, we provide existence of solution under the influence of the local geometry of $\G$ and the potential $h$.

math.AP↗

arXiv2015

Hardy and Hardy-Sobolev inequalities on Riemannian manifolds

Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \geq 3 $. Given $p_0 \in M$, $λ\in \mathcal{R}$ and $σ\in (0,2]$, we study existence and non existence of minimizers of the following quotient: \begin{equation}\label{Paper Equation} μ_{λ,σ}=\inf_{u \in H^1(M)\setminus \lbrace0\rbrace} \frac{\displaystyle\int_M |\nabla u|^2 dv_g -λ\int_M u^2 dv_g }{\biggl(\displaystyle\int_M ρ^{-σ} |u|^{2^*(σ)} dv_g\biggl)^{2/2^*(σ)}}, \end{equation} where $ρ(.):=dist(p_0,.)$ denoted the geodesic distance from $p \in M$ to $p_0$. In particular for $σ=2$, we provide sufficient and necessary conditions of existence of minimizers in terms of $λ$. For $σ\in (0,2)$ we prove existence of minimizers under scalar curvature pinching.

math.AP↗

arXiv2015

Weighted Hardy inequality on Riemannian manifolds

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$ and we let $Σ$ to be a closed submanifold of dimension $1 \leq k \leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $Σ$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers.

math.AP↗

arXiv2015

The role of the mean curvature in a Hardy-Sobolev trace inequality

The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

math.AP↗