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Mathématiques   > Accueil   > Avancées en Mathématiques Pures et Appliquées   > Numéro 2 (Spécial CSMT 2022)   > Article

Bifurcation au-delà des valeurs propres principales pour les problèmes de Neumann avec des poids indéfinis

Bifurcation beyond the principal eigenvalues for Neumann problems with indefinite weights


Marta Calanchi
Università degli Studi di Milano
Italy

Bernhard Ruf
Università degli Studi di Milano
Italy



Publié le 7 mars 2023   DOI : 10.21494/ISTE.OP.2023.0935

Résumé

Abstract

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This paper is devoted to the study of the effects of indefinite weights on the following nonlinear Neumann problems
$$$ {(P^\pm)} \begin{cases} -\Delta u &= \lambda \, a(x) u \pm |u|^{p-1}u\ &\quad \hbox{in } \Omega \subset ℝ^N \\ \frac{\partial u}{\partial \nu} &=\ 0 \ & \hbox{on } \partial \Omega \end{cases}$$$
The function $$$ a = a(x)$$$ is assumed to be continuous and sign-changing. Then the linear part has two sequences of eigenvalues. Our results establish a relation between the position of the parameter $$$\lambda$$$ and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem.

This paper is devoted to the study of the effects of indefinite weights on the following nonlinear Neumann problems
$$$ {(P^\pm)} \begin{cases} -\Delta u &= \lambda \, a(x) u \pm |u|^{p-1}u\ &\quad \hbox{in } \Omega \subset ℝ^N \\ \frac{\partial u}{\partial \nu} &=\ 0 \ & \hbox{on } \partial \Omega \end{cases}$$$
The function $$$ a = a(x)$$$ is assumed to be continuous and sign-changing. Then the linear part has two sequences of eigenvalues. Our results establish a relation between the position of the parameter $$$\lambda$$$ and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem.

eigenvalues indefinite weight Neumann problems bifurcation

eigenvalues indefinite weight Neumann problems bifurcation