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

Abstract

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