Nodal bubble-tower solutions  to radial elliptic problems near criticality

Andrés Contreras; Manuel del Pino

doi:10.3934/dcds.2006.16.525

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2006, Volume 16, Issue 3: 525-539. Doi: 10.3934/dcds.2006.16.525

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Nodal bubble-tower solutions to radial elliptic problems near criticality

Andrés Contreras^1, and
Manuel del Pino^1,

1.
Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago

Received: January 31, 2006

Revised: May 31, 2006

Published: August 2006

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Abstract

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.

Keywords:

nearly critical problem,
sign-changing solutions.

Mathematics Subject Classification: Primary: 35J25, 35J20; Secondary: 35B33.

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