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Multiple bubbling for the exponential nonlinearity in the slightly supercritical case

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  • Let $B$ denote the unit ball in $\mathbb R^2$. We consider the slightly super-critical Gelfand problem for the $p$-Laplacian operator $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$,

    $ -\Delta_{2-\varepsilon} u=\lambdae^u$ in $B\quad u =0 $ on $ \partial B,$

    for small $\varepsilon>0$. We show that if $k\ge 1$ is given and $\lambda>0$ is fixed and small, then there is a family of radial solutions exhibiting multiple blow-up as $\varepsilon\to 0$ in the form of a superposition of $k$ bubbles of different blow-up orders and shapes. Similar phenomena is found for the same problem involving the operator $\Delta_{N-\varepsilon}$ in $\mathbb R^N$, $N\ge 3$.

    Mathematics Subject Classification: Primary:35B25, 35B40; Secondary: 34C23, 35B32, 35B33, 35J60, 35J65, 35P30.


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