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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 2 (Special CSMT 2022)   > Article

Bifurcation beyond the principal eigenvalues for Neumann problems with indefinite weights

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


Marta Calanchi
Università degli Studi di Milano
Italy

Bernhard Ruf
Università degli Studi di Milano
Italy



Published on 7 March 2023   DOI : 10.21494/ISTE.OP.2023.0935

Abstract

Résumé

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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