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.
Categories: math.AP