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Topological Methods in Nonlinear Analysis

On global bifurcation for the nonlinear Steklov problems
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On global bifurcation for the nonlinear Steklov problems

Authors

  • Thazhe Veetil Anoop https://orcid.org/0000-0002-2470-9140
  • Nirjan Biswas https://orcid.org/0000-0002-3528-8388

DOI:

https://doi.org/10.12775/TMNA.2020.080

Keywords

Bifurcation, Steklov eigenvalue problem, weighted trace inequalities, Lorentz and Lorentz-Zygmund spaces

Abstract

For $p \in (1, \infty)$, for an integer $N \geq 2$ and for a bounded Lipschitz domain $\Omega \subset \R^N$, we consider the following nonlinear Steklov bifurcation problem $$ -\Delta_p \phi = 0 \quad \text{in } \Omega, \qquad |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} = \lambda \big( g \abs{\phi}^{p-2}\phi + f r(\phi) \big) \quad \text{on } \partial \Omega, $$ where $\Delta_p$ is the $p$-Laplace operator, $g,f \in L^1(\partial \Omega)$ are indefinite weight functions and $r \in C(\mathbb R)$ satisfies $r(0)=0$ and certain growth conditions near zero and at infinity. For $f$, $g$ in some appropriate Lorentz-Zygmund spaces, we establish the existence of a continuum that bifurcates from $(\lambda_1,0)$, where $\lambda_1$ is the first eigenvalue of the following nonlinear Steklov eigenvalue problem $$ -\Delta_p \phi = 0 \quad \text{in } \Omega, \qquad |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} = \lambda g \abs{\phi}^{p-2}\phi \quad \text{on } \partial \Omega. $$

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Published

2021-12-05

How to Cite

1.
ANOOP, Thazhe Veetil and BISWAS, Nirjan. On global bifurcation for the nonlinear Steklov problems. Topological Methods in Nonlinear Analysis. Online. 5 December 2021. Vol. 58, no. 2, pp. 731 - 763. [Accessed 18 May 2025]. DOI 10.12775/TMNA.2020.080.
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Vol 58, No 2 (December 2021)

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Copyright (c) 2021 Thazhe Veetil Anoop, Nirjan Biswas

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