A Doubly Critical Elliptic Problem with Submanifold Singularities

TL;DR

This paper investigates the existence of positive solutions to a doubly critical elliptic PDE with submanifold singularities, employing variational methods and analyzing the influence of geometry and potential near the singularity.

Contribution

It introduces new existence results for a doubly critical elliptic problem with submanifold singularities, highlighting the role of geometry and potential behavior.

Findings

Existence of positive solutions depends on the local geometry of the submanifold.

The potential function's behavior near the submanifold influences solution existence.

Variational methods and test functions are effective for doubly critical problems.

Abstract

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.