A sharp inequality of Trudinger-Moser type and extremal functions in H-1,H-n (R-n)

作者：do O Joao Marcos^*; de Souza Manasses

来源：Journal of Differential Equations, 2015, 258(11): 4062-4101.

DOI：10.1016/j.jde.2015.01.026

• Limiting Sobolev inequalities
• Trudinger-Moser inequality
• Sharp constants
• Extremal function
• Blow-up analysis

We prove a sharp form of the Trudinger-Moser inequality for the Sobolev space H-1,H-n (R-n). The sharpness comes from the introduction of an extra factor parallel to u parallel to(n)(n) in the classical Trudinger-Moser inequality. Let
l(alpha):= sup({u is an element of H1,n(Rn):parallel to u parallel to 1,n = 1}) integral(Rn) (Phi o nu alpha(u)dx,)
where Phi (t) := e(t) -Sigma(n-1)(i=0) t(i)/i(i) and nu(alpha) (u):= beta(n)(1 + alpha parallel to u parallel to(n)(n))(1/n-1))vertical bar u vertical bar(n/(n-1)). The main results read: (1) for 0 <= alpha < 1 we have l (alpha) < infinity, (2) for alpha > 1, l(alpha) = infinity and (3) moreover, we prove that for 0 <= alpha < 1, an extremal function for l(alpha) exists.

• 出版日期2015-6-5