Beyond the Trudinger-Moser supremum
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Date
2012
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Abstract
Let Omega be a bounded, smooth domain in R-2. We consider the functional
I(u) = integral(Omega)e(u2) dx
in the supercritical Trudinger-Moser regime, i.e. for integral(Omega)|del u|(2)dx > 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 > 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.
I(u) = integral(Omega)e(u2) dx
in the supercritical Trudinger-Moser regime, i.e. for integral(Omega)|del u|(2)dx > 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 > 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.