doi: 10.3934/dcds.2021154
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Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received  April 2021 Early access October 2021

Fund Project: Partially supported by the NNSF 11271044 of China

This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.

Citation: Guangcun Lu. Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021154
References:
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A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC Research Notes in Mathematics, 425. Chapman & Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

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T. Bartsch, Topological Methods for Variational Problems with Symmetries, Lecture Notes in Mathematics, 1560. Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0073859.  Google Scholar

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R. G. Bettiol, P. Piccione and G. Siciliano, Equivariant bifurcation in geometric variational problems, Analysis and Topology in Nonlinear Differential Equations, 103–133, Progr. Nonlinear Differential Equations Appl., 85, Birkhüser/Springer, Cham, 2014.  Google Scholar

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R. Böhme, Die Lösung der Verzweigungsgleichung für nichtlineare Eigenwertprobleme, Math. Z., 127 (1972), 105-126.  doi: 10.1007/BF01112603.  Google Scholar

[10]

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J. L. Dalec'kiǏ and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, 1974.  Google Scholar

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M. Degiovanni and M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl., 167 (1994), 73–100. doi: 10.1007/BF01760329.  Google Scholar

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[27]

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[28]

M. Field, Lectures on Bifurcations, Dynamics and Symmetry, Pitman Research Notes in Mathematics, 356., Chapman and Hall/CRC; 1996.  Google Scholar

[29]

P. M. Fitzpatrick and J. Pejsachowicz, Parity and generalized multiplicity, Transactions of the American Mathematical Society, 326 (1991), 281-305.  doi: 10.1090/S0002-9947-1991-1030507-7.  Google Scholar

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A. Ioffe and E. Schwartzman, An extension of the Rabinowitz bifurcation theorem to Lipschitz potenzial operators in Hilbert spaces, Proc. Amer. Math. Soc., 125 (1997), 2725-2732.  doi: 10.1090/S0002-9939-97-04061-6.  Google Scholar

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Y. Jabri, The Mountain Pass Theorem: Variants, Generalizations and Some Applications, Cambridge University Press, Cambridge 2003. doi: 10.1017/CBO9780511546655.  Google Scholar

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M. Jiang, A generalization of Morse lemma and its applications, Nonlinear Analysis, 36 (1999), 943-960.  doi: 10.1016/S0362-546X(97)00701-3.  Google Scholar

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A. Liapunov, Sur les figures d'equilibrium, Acad. Nauk St. Petersberg, (1906), 1–225. Google Scholar

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J. Q. Liu, Bifurcation for potential operators, Nonlinear Anal., 15 (1990), 345-353.  doi: 10.1016/0362-546X(90)90143-5.  Google Scholar

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G. Lu, Corrigendum to "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems" [J. Funct. Anal., 256 (2009), 2967–3034], J. Funct. Anal., 261 (2011), 542-589.  doi: 10.1016/j.jfa.2011.02.027. J.+Funct.+Anal.,+256+(2009),+2967–3034]+G. Lu+2011" target="_new" title="Go to article in Google Scholar"> Google Scholar

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[43]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces III. The case of critical manifolds, Journal Nonlinear Analysis and Application, 2019, 41–63. doi: 10.5899/2019/jnaa-00337.  Google Scholar

[44]

G. Lu, Splitting lemmas for the Finsler energy functional on the space of $H^1$-curves, Proc. London Math. Soc., $\textsf {113}$ (2016), 24–76. doi: 10.1112/plms/pdw022.  Google Scholar

[45]

G. Lu, Morse theory methods for a class of quasi-linear elliptic systems of higher order, Calc. Var. Partial Differential Equations, 58 (2019), Art. 134, 49 pp. doi: 10.1007/s00526-019-1577-1.  Google Scholar

[46]

G. Lu, Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems, Discrete Contin. Dyn. Syst., (2021). doi: 10.3934/dcds.2021155.  Google Scholar

[47]

G. Lu, Variational methods for Lagrangian systems of higher order, In Progress. Google Scholar

[48]

A. Marino, La biforcazione nel caso variazionale, (Italian), Confer. Sem. Mat. Univ. Bari No. 132(1973), 14 pp.  Google Scholar

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