A Doubly Critical Elliptic Problem with Submanifold Singularities

Let $N \ge 4$, $\Omega$ be a bounded domain in $\mathbb{R}^N$, and let $\Sigma \subset \Omega$ be a smooth closed submanifold of dimension $k$ with $2 \le k \le N-2$. We study the existence of positive solutions $u \in H_0^1(\Omega)$ to the Euler--Lagrange equation $$-\Delta u + h u = \lambda\, \rho_{\Sigma}^{-s_1}\, u^{2^{*}_{s_1}-1} + \rho_{\Sigma}^{-s_2}\, u^{2^{*}_{s_2}-1} \quad \text{in } \Omega,$$ where $h : \Omega \to \mathbb{R}$ is a continuous potential, $\lambda > 0$ 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 $\rho_{\Sigma} = \mathrm{dist}(\,\cdot\,, \Sigma)$ denotes the distance to the submanifold $\Sigma$. 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 $\Sigma$ and the behavior of the potential $h$ near $\Sigma$ play a crucial role in the existence of positive solutions for this doubly critical problem.