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Trudinger–Moser inequality on the whole plane with the exact growth condition

Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi (2015)

Journal of the European Mathematical Society

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function....

Two blow-up regimes for L 2 supercritical nonlinear Schrödinger equations

Frank Merle, Pierre Raphaël, Jérémie Szeftel (2009/2010)

Séminaire Équations aux dérivées partielles

We consider the focusing nonlinear Schrödinger equations i t u + Δ u + u | u | p - 1 = 0 . We prove the existence of two finite time blow up dynamics in the supercritical case and provide for each a qualitative description of the singularity formation near the blow up time.

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