[FORTHCOMING] Hardy-Sobolev critical equations with totally geodesic singularities: existence via the mountain pass theorem

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).

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).

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