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ON THE BEHAVIOUR OF THE FIRST EIGENFUNCTION OF THE $p$-LAPLACIAN NEAR ITS CRITICAL POINTS

Published online by Cambridge University Press:  12 May 2003

JORGE GARCÍA-MELIÁN
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna 38271 La Laguna, Tenerife (Spain) jjgarmel@ull.es
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Abstract

In this paper, the behaviour of the positive eigenfunction $\phi$ of $\Lp u=\la |u|^{p-2}u$ in $\Om$, $u_{|\p \Om} =0$, $p>1$, is studied near its critical points. Under some convexity and symmetry assumptions on $\Om$, $\phi$ is seen to have a unique critical point at $x=0$; also, the behaviour of both $\phi$ and $\nabla\phi$ is determined nearby. Positive solutions $u$ to some general problems $\Lp u=f(u)$ in $\Om$, $u_{|\p \Om} =0$, are also considered, with some convexity restrictions on $u$.

Type
Research Article
Copyright
© The London Mathematical Society 2003

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Footnotes

Supported by a MCYT project under contract #BFM2001-3894.