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September  2006, 16(3): 525-539. doi: 10.3934/dcds.2006.16.525

Nodal bubble-tower solutions to radial elliptic problems near criticality

1. 

Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile, Chile

Received  February 2006 Revised  June 2006 Published  August 2006

We describe as $\varepsilon \to 0$ radially symmetric sign-changing solutions to the problem

$ -\Delta u =|u|^{\frac 4{N-2} -\varepsilon} u \quad \text{in } B $

where $B$ is the unit ball in $\R^N$, $N\ge 3$, under zero Dirichlet boundary conditions. We construct radial solutions with $k$ nodal regions which resemble a superposition of "bubbles'' of different signs and blow-up orders, concentrating around the origin. A dual phenomenon is described for the slightly supercritical problem

$ -\Delta u =|u|^{\frac 4{N-2} +\varepsilon} u \quad \text{in } \R^N \setminus B $

under Dirichlet and fast vanishing-at-infinity conditions.

Citation: Andrés Contreras, Manuel del Pino. Nodal bubble-tower solutions to radial elliptic problems near criticality. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 525-539. doi: 10.3934/dcds.2006.16.525
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