A Doubly Critical Elliptic Problem with Submanifold Singularities

Abstract

Let $N\ge 4$, $\hat{I}\text{c◯}$ be a bounded domain in ${\mathbb{R}}^N$, and let $\hat{I}\pounds \subset \hat{I}\text{c◯}$ be a smooth closed submanifold of dimension $k$ with $2\le k\le N-2$. We study the existence of positive solutions $u\in H_0^1(\hat{I}\text{c◯})$ 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 $h:\hat{I}\text{c◯}\to \mathbb{R}$ is a continuous potential, $\hat{I}»>0$ 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 $\ddot{I}{}_{\hat{I}}\pounds =\mathrm{dist}(\cdot ,\hat{I}\pounds )$ denotes the distance to the submanifold $\hat{I}\pounds$. 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 $\hat{I}\pounds$ and the behavior of the potential $h$ near $\hat{I}\pounds$ play a crucial role in the existence of positive solutions for this doubly critical problem.