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

Abstract

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schrödinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$-\text{div}(|x{|}^{-2a}\nabla u)=\hat{I}»\frac{u}{|x{|}^{2a}}+{\hat{I}}^2\frac{|u{|}^{q-2}u}{|x{|}^{bq}}+\frac{|u{|}^{2^{♯}-2}u}{|x{|}^{b{2}^{♯}}}\text{in}{\mathbb{R}}^N,$$ $${\int }_{{\mathbb{R}}^N}\frac{|u{|}^2}{|x{|}^{2a}}dx=\ddot{I}{}^2,$$ where $\hat{I}»\in \mathbb{R}$, ${\hat{I}}^2,\ddot{I}>0$, $0<a<\frac{N-2}{2}$, $a<b<a+1$, $2^{♯}:=\frac{2N}{N-2(1+a-b)}$ and $2<q<2^{♯}$. 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.