Seminar Analysis: Luca Martinazzi (Rutgers)

The Moser-Trudinger equation on a disk: blow-up behavior and non-existence

We study the Moser-Trudinger equation Δu = λu Exp(u2), λ>0 on a 2-dimensional disk, arising from the Moser-Trudinger sharp embedding of H¹₀(Disk) into the Orlicz space of functions u with Exp(u2) integrable. We answer some long standing open questions:

a) The weak limit of a blowing-up sequence of solutions to the Moser-Trudinger equation on a disk is 0.

b) The Dirichlet energy of a blowing-up sequence of solutions on a disk converges to 4π.

c) For L large enough, the Moser-Trudinger equation on a disk admits no solution with Dirichlet energy larger than L.

This work is joint project with Andrea Malchiodi (SISSA - Trieste).

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