Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR 2

Let Ω be a bounded smooth domain inR ². Letf:R→R be a smooth non-linearity behaving like exp{s ²} ass→∞. LetF denote the primitive off. Consider the functionalJ:H ₀ ¹ (Ω)→R given by It can be shown thatJ is the energy functional associated to the following nonlinear problem: −Δu=f(u) in Ω,u=0 on ρΩ. In this paper we consider the global compactness properties ofJ. We prove thatJ fails to satisfy the Palais-Smale condition at the energy levels {k/2},k any positive integer. More interestingly, we show thatJ fails to satisfy the Palais-Smale condition at these energy levels along two Palais-Smale sequences. These two sequences exhibit different blow-up behaviours. This is in sharp contrast to the situation in higher dimensions where there is essentially one Palais-Smale sequence for the corresponding energy functional.