Multiple bubbling for the exponential nonlinearity in the slightly supercritical case

Manuel del Pino; Jean Dolbeault; Monica Musso

doi:10.3934/cpaa.2006.5.463

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2006, Volume 5, Issue 3: 463-482. Doi: 10.3934/cpaa.2006.5.463

This issue Previous Article Stable determination of a semilinear term in a parabolic equation Next Article The study on solutions to Camassa-Holm equation with weak dissipation

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
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: December 31, 2004

Revised: March 31, 2006

Published: August 2006

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Abstract

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

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

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