Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR
2 |
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Authors: | A Dimurthi S Prashanth |
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Institution: | (1) T.I.F.R. Centre, P.O. Box No. 1234, 560 012 Bangalore, India |
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Abstract: | 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. |
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Keywords: | Blow-up analysis critical exponent problem inR 2 Moser functions Palais-Smale sequence Palais-Smale condition |
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