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

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 $u\in H_0^1\left(\Omega \right)$ to the equation $$-\Delta u+hu=\lambda {\rho }_{\Gamma }^{-s_1}{u}^{5-2s_1}+{\rho }_{\Gamma }^{-s_2}{u}^{5-2s_2}\text{ in }\Omega$$ 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 $-\Delta +h$.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1702.02202