• Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR2

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      https://www.ias.ac.in/article/fulltext/pmsc/107/03/0283-0317

    • Keywords

       

      Blow-up analysis; critical exponent problem inR2; Moser functions; Palais-Smale sequence; Palais-Smale condition

    • Abstract

       

      Let Ω be a bounded smooth domain inR2. Letf:RR be a smooth non-linearity behaving like exp{s2} ass→∞. LetF denote the primitive off. Consider the functionalJ:H01(Ω)→R given by$$J(u) = \frac{1}{2}\int_\Omega {\left| {\nabla u} \right|^2 dx - } \int_\Omega {F(u)dx.} $$ 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.

    • Author Affiliations

       

      A Dimurthi1 S Prashanth1

      1. T.I.F.R. Centre, P.O. Box No. 1234, Bangalore - 560 012, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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