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

• # Fulltext

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

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

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019