Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation

2018/01/11 by Csato, Gyula, Roy, Prosenjit, Nguyen, Van Hoang · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.03932

Abstract

The Moser-Trudinger embedding has been generalized in [Adimurthi A.; Sandeep K., A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), no. 5-6, 585--603] to the following weighted version: if Ω⊂ℝⁿ is bounded, ω_n-1 is the H^n-1 measure of the unit sphere, then for α>0 and β∈ [0,n), sup_u\inB1∫_Ω\frace^α|u|^n/(n-1)|x|^β≤ C ⇔ \fracαα_n+\fracβn≤1, where α_n=n\cnn and B₁ = \ u ∈ W₀^{1, n}(Ω) | ∫_Ω |∇ u |ⁿ ≤1 \. We prove that the supremum is attained on any domain Ω. The paper also fills in the gaps in the proof of [Lin K.C., Extremal functions for Moser's inequality, Trans. of. Am. Math. Soc., 384 (1996), 2663--2671], which deals with the case β=0.

Cited by

• Compactness of Extremals for Singular Anisotropic Trudinger-Moser functionals on bounded domain
• Existence of extremal functions and Wulff symmetry for anisotropic Trudinger-Moser inequalities

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