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

Abstract

We let $Ω$ be a bounded domain of $\mathbb{R}^3$ and $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the equation $$ -Δu+hu=λρ^{-s_1}_Γu^{5-2s_1}+ρ^{-s_2}_Γu^{5-2s_2} \qquad \textrm{ in } Ω$$ where $h$ is a continuous function and $ρ_Γ$ is the distance function to $Γ$. 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 $-Δ+h$.