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