# American Institute of Mathematical Sciences

September  2006, 5(3): 463-482. doi: 10.3934/cpaa.2006.5.463

## Multiple bubbling for the exponential nonlinearity in the slightly supercritical case

 1 Departamento de Ingeneria Matematica F.C.F.M., Casilla 170 Correro 3, Santiago, Chile 2 Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16 3 Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino

Received  January 2005 Revised  April 2006 Published  June 2006

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 $\mathbb R^N$, $N\ge 3$.

Citation: Manuel del Pino, Jean Dolbeault, Monica Musso. Multiple bubbling for the exponential nonlinearity in the slightly supercritical case. Communications on Pure and Applied Analysis, 2006, 5 (3) : 463-482. doi: 10.3934/cpaa.2006.5.463
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