On the Neumann Problem of Hardy-Sobolev critical equations with the multiple singularities

TL;DR

This paper investigates the existence of positive solutions for Hardy-Sobolev critical equations with multiple singularities in bounded domains, establishing conditions under which solutions exist depending on the location of singularities and parameters.

Contribution

It extends the existence theory of solutions to Hardy-Sobolev equations with multiple singularities, including boundary singularities with positive curvature and varying exponents.

Findings

Positive solutions exist for small positive mbda.

Solutions exist for boundary singularities with positive mean curvature for any mbda > 0.

Extended existence results for multiple singularities with different exponents.

Abstract

Let $N \geq 3$ and $\Omega \subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(\Omega)$ of \begin{align*} \left\{ \begin{array}{l} -\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\text{ in }\Omega\\ \frac{\partial u}{\partial \nu} = 0 \text{ on }\partial\Omega, \end{array}\right. \end{align*} where $0 < s <2$, $2^*(s) = \frac{2(N-s)}{N-2}$ and $x_1, x_2 \in \overline{\Omega}$ with $x_1 \neq x_2$. First, we show the existence of positive solutions to the equation provided the positive $\lambda$ is small enough. In case that one of the singularities locates on the boundary and the mean curvature of the boundary at this singularity is positive, the existence of positive solutions is always obtained for any $\lambda > 0$. Furthermore, we extend the existence theory of solutions to the equations for the case of the multiple singularities with different exponents.