Normalized Solutions for a Weighted Laplacian Problem with the Caffarelli-Kohn-Nirenberg Critical Exponent

TL;DR

This paper proves the existence and multiplicity of normalized solutions for a weighted nonlinear Schrödinger equation involving the Caffarelli-Kohn-Nirenberg operator, using advanced variational methods and refined estimates.

Contribution

It introduces new techniques to establish multiple solutions for a weighted Schrödinger equation with critical exponents, addressing noncompactness challenges.

Findings

Existence of mass-subcritical ground states.

Multiple constrained critical points found.

High-energy solutions in critical regimes.

Abstract

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schr\"odinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$ -\text{div}(|x|^{-2a}\nabla u)=\lambda \frac{u}{|x|^{2a}}+\beta\frac{|u|^{q-2}u}{|x|^{bq}} +\frac{|u|^{2^{\sharp}-2}u}{|x|^{b{2^{\sharp}}}}\quad \text{in}~\mathbb{R}^N,$$ $$\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}}dx=\rho^2,$$ where $\lambda\in \mathbb{R}$, $\beta,~\rho>0$, $0< a<\frac{N-2}{2}$, $a<b<a+1$, $2^{\sharp}:=\frac{2N}{N-2(1+a-b)}$ and $2<q<{2^{\sharp}}$. Through constrained variational techniques, refined estimates on the best constants in the Caffarelli-Kohn-Nirenberg inequalities, and a bespoke concentration-compactness lemma, the study secures mass-subcritical ground states alongside multiple constrained critical points, together with high-energy ground state solutions in the mass-critical and supercritical regimes -- notwithstanding the noncompactness arising from the critical Caffarelli-Kohn-Nirenberg nonlinearity over the unbounded domain.