TY  - Type of reference 
TI  - [FORTHCOMING] Équations critiques de Hardy–Sobolev avec singularités totalement géodésiques : existence par le théorème du col de montagne. 
AU  - El Hadji Abdoulaye Thiam 
AB  - We consider a compact Riemannian manifold $$$(M, g)$$$ of dimension $$$N \geq 3$$$ and $$$\Sigma$$$ a closed totally geodesic submanifold of dimension $$$1 \leq k \leq N-2$$$, and $$$h: M \to ℝ$$$ is a continuous function such that the linear operator $$$-Δ_g+h$$$ is coercive. We study existence of positive solutions $$$u \in H^1\left(M\right)$$$ to the following nonlinear PDE with two Hardy-Sobolev critical exponents :
(0.1)  $$$ -\Delta_g u+h u=\lambda \rho_{\Sigma}^{-s_1} u^{2^*_{s_1}-1}+\rho_{\Sigma}^{-s_2} u^{2^*_{s_2}-1} \qquad \textrm{ in } (M, g)$$$
where $$$\lambda$$$ is a positive parameter, $$$0 < s_2 < s_1 < 2$$$, the $$$2^*_{s_i}:=\frac{2(N-s_i)}{N-2}$$$ $$$(i=1, 2)$$$ are two critical Hardy-Sobolev exponents and $$$\rho_\Sigma: \mathcal{M} \to ℝ$$$ is the distance function to $$$\Sigma$$$. In this paper, we give sufficient condition depending on the local geometries of the submanifold $$$\Sigma$$$ and the manifold $$$M$$$, for the existence of mountain pass solution to (0.1). 
DO  - À venir 
JF  - Avancées en Mathématiques Pures et Appliquées 
KW  - Mountain Pass Solution, Two Hardy-Sobolev critical exponents, Scalar Curvature, Riemannian curvature tensor, Submanifold, Mountain Pass Solution, Two Hardy-Sobolev critical exponents, Scalar Curvature, Riemannian curvature tensor, Submanifold,  
L1  -  
LA  - fr 
PB  - ISTE OpenScience 
DA  - 2026/06/19 
SN  - 1869-6090 
TT  - [FORTHCOMING] Hardy-Sobolev critical equations with totally geodesic singularities: existence via the mountain pass theorem 
UR  - https://openscience.fr/Equations-critiques-de-Hardy-Sobolev-avec-singularites-totalement-geodesiques 
IS  - Articles à paraître<br> 
VL  -  
ER  -