Normalized Solutions Achieved for Weighted Laplacian Problem with Exponent
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Scientists have long sought to understand the behaviour of nonlinear Schrödinger-type equations, and a new study by Divya Goel and Asmita Rai establishes the existence of normalized solutions for a weighted Laplacian problem with the Caffarelli-Kohn-Nirenberg critical exponent. Their work addresses a particularly challenging scenario involving critical and supercritical regimes where noncompactness typically prevents the discovery of stable solutions. By employing constrained variational techniques and developing a new concentration-compactness lemma, Goel and Rai secure not only mass-subcritical ground states but also multiple constrained critical points and high-energy ground state solutions, significantly advancing our knowledge of this complex mathematical problem and potentially informing future research in areas like fluid dynamics and quantum mechanics. Their work addresses a particularly challenging scenario involving critical and supercritical regimes where noncompactness typically prevents the discovery of stable solutions. Weighted nonlinear Schrödinger equations and normalised solutions Here, λ is a real number, β and ρ are positive, 0 a N−2 2, a b a + 1, 2♯:= 2N N−2(1+a−b), and 2 q 2♯. This is accomplished by leveraging the constrained variational methods, which allow for the investigation of solutions subject to the prescribed mass constraint. The refined estimates on the Caffarelli-Kohn-Nirenberg inequalities are crucial for controlling the behaviour of the solutions and ensuring the validity of the concentration-compactness lemma. Experiments show that the concentration-compactness lemma, specifically tailored for this problem, plays a vital role in handling the singular nature of the equation and the lack of compactness in the solution space.
The team proved that the functional associated with the equation admits both minimizers and critical points, providing a comprehensive understanding of the solution structure. The work opens avenues for exploring the stability and orbital properties of these solutions, which are relevant to physical models in nonlinear optics and Bose-Einstein condensation. This establishes a clear separation between the different energy regimes and provides insights into the behaviour of the solutions as the mass approaches critical values. The research establishes a rigorous mathematical framework for understanding normalized solutions to this class of equations, paving the way for applications in diverse areas of physics and engineering.
The team’s findings contribute significantly to the broader field of nonlinear partial differential equations and provide valuable tools for analysing similar problems with singular potentials and critical exponents.
Constrained Variational Analysis of Weighted Schrödinger Equations offers This work builds upon the foundational Caffarelli, Kohn, and Nirenberg inequalities, extending Sobolev, Hardy, and Gagliardo-Nirenberg inequalities to govern a functional framework for singular and weighted partial differential equations on RN. Researchers meticulously analysed this prototypical example, coupling the CKN operator with both subcritical and critical weighted nonlinearities to explore analytical difficulties associated with singular potentials, critical exponents, and a lack of compactness.
The team developed a bespoke concentration-compactness lemma, crucial for handling the noncompactness inherent in the critical nonlinearity over the unbounded domain. This constraint is central to modelling physical phenomena like nonlinear optics and Bose-Einstein condensation, where mass conservation is paramount. The approach leverages a norm-constrained variational framework, initially proposed by Cazenave and Lions, to investigate existence and orbital stability in the mass-subcritical setting via constrained minimization. Experiments employed a detailed analysis of the energy functional, distinguishing between mass-subcritical, mass-critical, and mass-supercritical regimes based on the exponent p. Specifically, when 2 2 + 4 N, the energy is unbounded, necessitating a variational scheme with mountain pass geometry, combined with compactness tools, to obtain normalized solutions. The research extended this analysis to equations with combined power-type nonlinearities, −∆u = λu + α|u|q−2u + |u|p−2u in RN, under the constraint ∥u∥2 2 = ρ2 with 2 q p ≤2∗:= 2N N−2, providing a comprehensive understanding of solution behaviour across different parameter regimes. Multiple solutions for weighted nonlinear Schrödinger equations exist The study addresses analytical difficulties associated with singular potentials, critical exponents, and a lack of compactness. This constraint fixes the “mass” of the state under consideration, a concept crucial in physical models like nonlinear optics and Bose, Einstein condensation where mass conservation is fundamental. Measurements confirm that the functional associated with the equation is coercive and bounded from below in the mass-subcritical case, allowing for the direct variational minimization and identification of a global minimizer. Data shows that when considering the mass-supercritical case, the energy becomes unbounded from below along the constraint, necessitating a more delicate approach. Tests prove the construction of a variational scheme with mountain pass geometry on the Pohozaev manifold, combined with suitable compactness tools, to obtain normalized solutions. Measurements of the Caffarelli-Kohn-Nirenberg inequality, a blend of Sobolev and Hardy inequalities, provide a foundation for this work, establishing the constant Ca,b = Ca,b(N, ) for all functions u in C∞0(RN), where ∥|x|γu∥q ≤Ca,b ∥|x|α∇u∥δ p ∥|x|ηu∥1−δ r. This work opens avenues for exploring new classes of inequalities and their applications in diverse physical theories.
Multiple Normalised Solutions via Caffarelli-Kohn-Nirenberg Analysis The study builds upon the Caffarelli-Kohn-Nirenberg inequality, a theorem blending Sobolev and Hardy inequalities with flexible weights and exponents, to investigate the equation (Pa ρ). By applying this inequality with specific exponent transformations, the authors analysed the problem across three cases: mass-subcritical (q qc). The authors acknowledge that their compactness results rely on specific conditions regarding the parameters a and b, and the embedding of function spaces is limited by these constraints. Future research could explore the behaviour of solutions for different parameter values or investigate the stability of the obtained ground states. 👉 More information 🗞 Normalized Solutions for a Weighted Laplacian Problem with the Caffarelli-Kohn-Nirenberg Critical Exponent 🧠 ArXiv: https://arxiv.org/abs/2601.20513 Tags: