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 $$-\Delta u + \lambda u = u^{-r} + u^{p-1} \quad \text{in } \Omega,$$ with the Dirichlet boundary condition $u=0$ on $\partial\Omega$ and the normalization constraint $\int_\Omega u^2\,dx = \rho$. Here $\Omega\subset\mathbb{R}^N$ ($N\ge3$) is a smooth bounded domain, $0<r<1$, $2+\frac{4}{N}<p<2^*$, where $2^*$ 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 $(\lambda,u)\in\mathbb{R}\times H_0^1(\Omega)$. The proof is based on a variational approach using a regularized functional and a careful analysis of the limiting process.