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

TL;DR

This work addresses the existence of positive normalized solutions to a singular elliptic equation with a prescribed $L^2$-norm on a bounded domain. By introducing a regularized functional to handle the singular term and minimizing on a mass-constrained set, the authors derive uniform estimates, including an $L^∞$ bound via a blow-up analysis, and pass to the limit to obtain a weak solution of the original problem with a Lagrange multiplier $\lambda$. The main result proves existence for sufficiently small mass $\rho$, yielding a pair $(\lambda,u)$ with $u>0$ solving $-\Delta u+\lambda u = u^{-r}+u^{p-1}$ in $Ω$ and $u=0$ on $∂Ω$. The approach combines constrained variational methods, regularization, and careful limiting arguments to handle the interplay between the singular term and the $L^2$-constraint, extending the theory of normalized solutions to singular problems on bounded domains.

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 $\partialΩ$ 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.

Theorems & Definitions (23)

• Theorem 1.1
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof