Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem in R2

Adimurthi, . ; Prashanth, S. (1997) Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem in R2 Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 107 (3). pp. 283-317. ISSN 0253-4142

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Official URL: http://www-old.ias.ac.in/j_archive/mathsci/107/vol...

Related URL: http://dx.doi.org/10.1007/BF02867260

Abstract

Let Ω be a bounded smooth domain inR2. Letf:R→R 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)=12∫Ω|∇u|2dx−∫ΩF(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 that J 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.

Item Type:Article
Source:Copyright of this article belongs to Indian Academy of Sciences.
Keywords:Blow-Up Analysis; Critical Exponent Problem in R2; Moser Functions; Palais-Smale Sequence; Palais-Smale Condition
ID Code:112170
Deposited On:31 Jan 2018 04:32
Last Modified:31 Jan 2018 04:32

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