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

TL;DR

This paper proves the existence of positive normalized solutions for a singular elliptic equation with a supercritical nonlinearity, using variational methods and analysis of the limiting process for small prescribed mass.

Contribution

It establishes the existence of solutions for a class of singular elliptic equations with supercritical nonlinearities under normalization constraints, extending previous results to this challenging setting.

Findings

Positive solutions exist for small mass parameter  ho

Solutions are obtained via a variational approach with regularization

The analysis handles the supercritical nonlinearity and singular term

Abstract

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.