Beyond the Trudinger-Moser supremum

作者：del Pino Manuel; Musso Monica; Ruf Bernhard^*

来源：Calculus of Variations and Partial Differential Equations, 2012, 44(3-4): 543-576.

DOI：10.1007/s00526-011-0444-5

Let Omega be a bounded, smooth domain in R-2. We consider the functional %26lt;br%26gt;I(u) = integral(Omega)e(u2) dx %26lt;br%26gt;in the supercritical Trudinger-Moser regime, i.e. for integral(Omega)|del u|(2)dx %26gt; 4 pi. More precisely, we are looking for critical points of I(u) in the class of functions u is an element of H-0(1) (Omega) such that integral(Omega)|del u|(2)dx = 4 pi k (1+ alpha), for smalla alpha %26gt; 0. In particular, we prove the existence of 1-peak critical points of I(u) with integral(Omega)|del u|(2)dx = 4 pi(1 + alpha) for any bounded domain Omega, 2-peak critical points with integral(Omega)|del u|(2)dx = 8 pi(1 + alpha) for non-simply connected domains Omega, and k-peak critical points with integral(Omega)|del u|(2)dx = 4kp(1 + alpha) if Omega is an annulus.

• 出版日期2012-7