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Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains

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  • We construct solutions of the semilinear elliptic problem

    $\Delta u+ |u|^{p-1}u+$ε1/2 f = 0 in Ω
    u=ε1/2 g on $\partial$Ω

    in a bounded smooth domain $\Omega \subset \R^N$ $(N\geq 3)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$. As $p\rightarrow \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. As applications, we will give some existence results, in particular, when $\O$ are symmetric domains perforated with the small hole and when $f=0$ and $g=0$.

    Mathematics Subject Classification: 35J60, 35J25.


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