Positive normalized solutions to a singular elliptic equation with a $L^2$-supercritical nonlinearity

This paper studies the existence of positive normalized solutions to the singular elliptic equation $$-\mathrm{\Delta }u+\lambda u=u^{-r}+u^{p-1}\text{in }\mathrm{\Omega },$$ with the Dirichlet boundary condition $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$ and the normalization constraint ${\int }_{\mathrm{\Omega }}{u}^{2}dx=\rho$. Here $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) is a smooth bounded domain, $0<r<1$, $2+\frac{4}{N}<p<2^{\ast }$, where $2^{\ast }$ is the critical Sobolev exponent, and $\lambda \in \mathbb{R}$ is a Lagrange multiplier. We obtain that for sufficiently small $\rho >0$, the problem admits a positive solution $(\lambda ,u)\in \mathbb{R}\times H_0^1(\mathrm{\Omega })$. The proof is based on a variational approach using a regularized functional and a careful analysis of the limiting process.