The Brezis-Nirenberg problem for the Laplacian with a singular drift in ℝⁿ and \mathbbSⁿ.

2015/03/21 by Rafael D. Benguria, Benguria, Rafael D., Soledad Benguria +1
Computer Science · Mathematics · #35P30 (Primary) 35J60 (Secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1503.06347

openalex publication_date 2015/03/21 · openalex created_date 2019/03/22 · openalex updated_date 2026/07/28

Abstract

We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both ℝⁿ and \mathbbSⁿ, 3 ≤ n ≤ 5. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter (δ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space is characterized by the (geodesic) radius of the ball, δ, and λ, the coupling constant of the linear term of the Brezis-Nirenberg problem.

Citations

• A relation between pointwise convergence of functions and convergence of functionals

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