A Doubly Critical Elliptic Problem with Submanifold Singularities

Let $N\ge 4$, $\mathrm{\Omega }$ be a bounded domain in ${\mathbb{R}}^N$, and let $\mathrm{\Sigma }\subset \mathrm{\Omega }$ be a smooth closed submanifold of dimension $k$ with $2\le k\le N-2$. We study the existence of positive solutions $u\in H_0^1(\mathrm{\Omega })$ to the Euler–Lagrange equation $$-\mathrm{\Delta }u+hu=\lambda {\rho }_{\mathrm{\Sigma }}^{-s_1}{u}^{2_{s_1}^{\ast }-1}+{\rho }_{\mathrm{\Sigma }}^{-s_2}{u}^{2_{s_2}^{\ast }-1}\text{in }\mathrm{\Omega },$$ where $h:\mathrm{\Omega }\to \mathbb{R}$ is a continuous potential, $\lambda >0$ is a real parameter, and $0\le s_2<s_1<2$. For $i=1,2$, the exponents $$2_{s_i}^{\ast }=\frac{2(N-s_i)}{N-2}$$ correspond to Hardy–Sobolev critical growth, and ${\rho }_{\mathrm{\Sigma }}=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\cdot ,\mathrm{\Sigma })$ denotes the distance to the submanifold $\mathrm{\Sigma }$. 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 $\mathrm{\Sigma }$ and the behavior of the potential $h$ near $\mathrm{\Sigma }$ play a crucial role in the existence of positive solutions for this doubly critical problem.