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# On the Principal Eigencurve of the $p$-Laplacian: Stability Phenomena

Published:2006-09-01
Printed: Sep 2006
• Abdelouahed El Khalil
• Said El Manouni
• Mohammed Ouanan
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## Abstract

We show that each point of the principal eigencurve of the nonlinear problem $$-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega,$$ is stable (continuous) with respect to the exponent $p$ varying in $(1,\infty)$; we also prove some convergence results of the principal eigenfunctions corresponding.
 Keywords: $p$-Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stability
 MSC Classifications: 35P30 - Nonlinear eigenvalue problems, nonlinear spectral theory 35P60 - unknown classification 35P6035J70 - Degenerate elliptic equations

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