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<p>In this paper, we are concerned with the bifurcation problem for an elliptic system with singular nonlinearities. It is well-known that the classical Lyapunov-Schmidt reduction method is a powerful tool dealing with bifurcations for nonlinear equations when the linearized operator is Fredholm. However, the occurrence of singular perturbations may introduce the ‘small divisors’ and break this property, and the Nash-Moser iteration scheme is thus needed. In this paper we shall prove the existence of spatial-periodic solutions for a generalized steady Swift-Hohenberg equation with singular nonlinearities by combining the iteration scheme with the reduction.</p>

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Bifurcations For an Elliptic System with Singular Perturbations

  • 期刊论文

  • 发表日期：

    2025年8月25日

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  • Yingdu Dong,
  • Xiong Li

摘要

In this paper, we are concerned with the bifurcation problem for an elliptic system with singular nonlinearities. It is well-known that the classical Lyapunov-Schmidt reduction method is a powerful tool dealing with bifurcations for nonlinear equations when the linearized operator is Fredholm. However, the occurrence of singular perturbations may introduce the ‘small divisors’ and break this property, and the Nash-Moser iteration scheme is thus needed. In this paper we shall prove the existence of spatial-periodic solutions for a generalized steady Swift-Hohenberg equation with singular nonlinearities by combining the iteration scheme with the reduction.