Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents

TL;DR

This paper investigates the existence of positive solutions to a nonlinear elliptic PDE involving Hardy-Sobolev critical exponents, with solutions depending on the Green function's regular part and the mass parameter.

Contribution

It introduces new existence results for solutions based on the Green function's regular part and the mass in a PDE with Hardy-Sobolev critical exponents.

Findings

Existence of solutions depends on the Green function's regular part.

Positive mountain pass solutions are established.

Results connect the mass parameter to solution existence.

Abstract

We let $\Omega$ be a bounded domain of $\mathbb{R}^3$ and $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation $$ -\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{5-2s_1}+\rho^{-s_2}_\Gamma u^{5-2s_2} \qquad \textrm{ in } \Omega $$ where $h$ is a continuous function and $\rho_\Gamma$ is the distance function to $\Gamma$. We prove existence of solutions depending on the regular part of the Green function of linear operator. We prove the existence of positive mountain pass solutions for this Euler-Lagrange equation depending on the mass which is the regular part of the Green function of the linear operator $-\Delta+h$.