A deformation lemma for a Moser-Trudinger type functional

We consider the functional I(λ,u)=12∫Ω|∇u|2-λlog1|Ω|∫Ωeu,u∈H01(Ω).When λ=8πN (N any positive integer), I(λ,·) admits unbounded Palais-Smale sequences. To overcome this lack of compactness, we propose a new deformation lemma.