A sharpened Moser–Pohozaev–Trudinger inequality with mean value zero in R2R2

Let Ω⊂R2Ω⊂R2 be a smooth bounded domain, and q(x)q(x) be a polynomial with q(0)≠0q(0)≠0. Then under some hypothesis on q(x)q(x), there holds sup∫Ω|∇u|2dx=1,∫Ωudx=0∫Ωe2πu2q(0)q(∫Ωu2dx)dx<+∞. A sufficient condition will be given to assure that the above inequality does not hold. Furthermore, the existence of the extremal functions will be derived.