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<p>This work presents a comprehensive computational framework for studying bifurcation phenomena in nonlinear eigenvalue problems involving the <i>p</i>-Laplacian with nonlinear boundary conditions. We extend the theoretical analysis of Cuesta et al. (Electron J Differ Equ 2019(32):1–29, 2019) by developing a rigorous finite element method (FEM) formulation. The paper provides detailed derivations of the weak form, the discrete nonlinear problem, and the associated Newton iteration scheme. We establish a priori error estimates linking the discretization error to the regularity of the solution and implement path-following (continuation) techniques to trace the bifurcating branches. Extensive numerical experiments on two-dimensional domains confirm the theoretical predictions: bifurcation from both zero and infinity at the first eigenvalue <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and the splitting of the solution continuum into strictly positive and strictly negative branches.</p>

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Finite element analysis of bifurcation from the first eigenvalue of the p-Laplacian with nonlinear boundary conditions

  • 期刊论文

  • 发表日期：

    2026年5月7日

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  • Marcial Nguemfouo,
  • Pascaline Nshimirimana

摘要

This work presents a comprehensive computational framework for studying bifurcation phenomena in nonlinear eigenvalue problems involving the p-Laplacian with nonlinear boundary conditions. We extend the theoretical analysis of Cuesta et al. (Electron J Differ Equ 2019(32):1–29, 2019) by developing a rigorous finite element method (FEM) formulation. The paper provides detailed derivations of the weak form, the discrete nonlinear problem, and the associated Newton iteration scheme. We establish a priori error estimates linking the discretization error to the regularity of the solution and implement path-following (continuation) techniques to trace the bifurcating branches. Extensive numerical experiments on two-dimensional domains confirm the theoretical predictions: bifurcation from both zero and infinity at the first eigenvalue \(\lambda _1\) λ 1 and the splitting of the solution continuum into strictly positive and strictly negative branches.