arXiv subjects

Abdourahmane Diatta

Publications and source records attributed to Abdourahmane Diatta.

3 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↗