Remarks on the Moser-Trudinger inequality

We extend the Moser-Trudinger inequality[GRAPHICS]to any Euclidean domain satisfying Poincare's inequalitylambda(1) (Omega) := inf(0 not equivalent to u is an element of W01, N (Omega)) integral(Omega) vertical bar del u(x)vertical bar(N) dx/integral(Omega) vertical bar u(x)vertical bar(N) dx > 0.We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip Omega := R x (-1, 1).