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

TL;DR

The paper studies normalized solutions to a weighted nonlinear Schrödinger-type equation driven by the Caffarelli–Kohn–Nirenberg operator under a prescribed mass constraint. It develops a constrained variational framework on the mass sphere, employing a refined Pohozaev manifold structure and sharp CKN constants to obtain existence and multiplicity results across mass-subcritical, mass-critical, and mass-supercritical regimes, while addressing noncompactness via a tailored concentration-compactness argument. In the subcritical setting, it proves the existence of a mass-normalized ground state and a second, higher-energy solution; in the critical and supercritical ranges, it establishes the existence of ground states (under small coupling) and provides compactness results for Palais–Smale sequences. The analysis combines fibering maps, energy estimates, and appendix-derived asymptotics for the best constants, delivering a unified treatment of weighted normalized solutions with genuinely weighted exponents ($a,b\neq 0$).

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, $$ -\text{div}(|x|^{-2a}\nabla u)=λ\frac{u}{|x|^{2a}}+β\frac{|u|^{q-2}u}{|x|^{bq}} +\frac{|u|^{2^{\sharp}-2}u}{|x|^{b{2^{\sharp}}}}\quad \text{in}~\mathbb{R}^N,$$ $$\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}}dx=ρ^2,$$ where $λ\in \mathbb{R}$, $β,~ρ>0$, $0< a<\frac{N-2}{2}$, $a<b<a+1$, $2^{\sharp}:=\frac{2N}{N-2(1+a-b)}$ 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.

Figures (4)

• Figure : Figure 1: $N=3$
• Figure : Figure 3: $N \ge 5$
• Figure : Figure 1: $N=3$
• Figure : Figure 2: $N=4$

Theorems & Definitions (51)

• Theorem 1.1: Caffarelli-Kohn-Nirenberg inequality
• Proposition 2.1: Compact Radial Embedding
• proof
• Corollary 2.1
• Theorem 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Lemma 4.1
• proof