Potential Bifurcation Theorem for a Nonhomogeneous Operator Equation and Its Application

Guo Wei DAI, Ru Yun MA

Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (2) : 263-274.

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Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (2) : 263-274. DOI: 10.12386/A2022sxxb0020

Potential Bifurcation Theorem for a Nonhomogeneous Operator Equation and Its Application

  • Guo Wei DAI1, Ru Yun MA2
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Abstract

The bifurcation phenomenon of the operator equation λ(f1′(x)+f2′(x))=g1′(x)+g2′(x) is studied in this paper. Suppose f2′≡0, f1 and g1 are a-homogeneous, and some other suitable conditions hold, Fučík et al. obtained that each normalized LS-eigenvalue of λf1′(x)=g1′(x) is a bifurcation point of the operator equation above. This paper studies the inhomogeneous case of f1+f2. We establish the same results as theirs when f1, f2, g1 and g2 satisfy some suitable conditions. A Lyusternik-Shnirel'man theorem is obtained as a preliminary result. And for the application of our abstract theorems, the bifurcation phenomenon from arbitrary LS-eigenvalues is studied for a nonlocal elliptic problem.

Key words

Lyusternik-Shnirel'man theorem / potential bifurcation / nonlocal equation

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Guo Wei DAI, Ru Yun MA. Potential Bifurcation Theorem for a Nonhomogeneous Operator Equation and Its Application. Acta Mathematica Sinica, Chinese Series, 2022, 65(2): 263-274 https://doi.org/10.12386/A2022sxxb0020

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