A Trudinger-Moser type inequality and its extremal functions in dimension two

摘要

Let Ω be a smooth bounded domain in R2 , W1,20 (Ω) be the usual Sobolev space andλ (Ω) be the first eigenvalue of the Laplace-Beltrami operator, sayλ (Ω) = infu∈W1,20 (Ω), Ω u2dx=1 Ω|?u|2dx.Using blow-up analysis, we prove that for real numbers α λ (Ω) and β 4π , the supremumsupu∈W1,20 (Ω), Ω |?u|2dx?α Ω u2dx 1 Ω(e4πu2 ?β u2)dxcan be attained by some function u ∈W1,20 (Ω) with Ω|?u|2dx?α Ω u2dx = 1. In the caseβ = 0, this is reduced to a result of Yang [24].