| Abstract: |
| In this talk, we are concerned with the compactness of the embedding
$H_0^1(\Omega)\hookrightarrow L^{p(\cdot)}(\Omega)$, where $\Omega$ is a bounded
domain in ${\Bbb R}^N$ with $N\geq 3$ and $p(\cdot)$ is a function called
a variable exponent satisfying $p(0)=2^*=\frac{2N}{N-2}$, $p(\cdot)\leq
2^*$ in $\Omega\setminus\{0\}$. It is well known that the embedding above is continuous but fails to be compact if $p(\cdot)\equiv 2^*$ due to the action
of the dilation. In this talk, we give an ``almost necessary and sufficient``
condition on the decay rate of
$2^*-p(x)$ as $|x|\to 0$ which assures the compactness of (S). Our approach is based on the profile-decomposition
of bounded sequences in $H_0^1$ together with the scaling argument. Other related topics are also discussed. |
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