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

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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 10 (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.

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

    A Dimurthi & S Prashanth

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  1. A Dimurthi
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  2. S Prashanth
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Correspondence to A Dimurthi.

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Dimurthi, A., Prashanth, S. Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR 2 . Proc. Indian Acad. Sci. (Math. Sci.) 107, 283–317 (1997). https://doi.org/10.1007/BF02867260

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