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 [ -Δu + λu = u^{-r} + u^{p-1} \quad \text{in } Ω, ] with the Dirichlet boundary condition $u=0$ on $\partialΩ$ and the normalization constraint $\int_Ωu^2,dx = ρ$. Here $Ω\subset\mathbb{R}^N$ ($N\ge3$) is a smooth bounded domain, $0<r<1$, $2+\frac{4}{N}<p<2^$, where $2^$ is the critical Sobolev exponent, and $λ\in\mathbb{R}$ is a Lagrange multiplier. We obtain that for sufficiently small $ρ>0$, the problem admits a positive solution $(λ,u)\in\mathbb{R}\times H_0^1(Ω)$. The proof is based on a variational approach using a regularized functional and a careful analysis of the limiting process.

💡 Research Summary

The paper investigates the Dirichlet problem
\