• Previous Article
    Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems
  • DCDS Home
  • This Issue
  • Next Article
    Statistics of multipliers for hyperbolic rational maps
March  2022, 42(3): 1243-1316. doi: 10.3934/dcds.2021154

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 Published  March 2022 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 and Continuous Dynamical Systems, 2022, 42 (3) : 1243-1316. doi: 10.3934/dcds.2021154
References:
[1]

A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC Research Notes in Mathematics, 425. Chapman & Hall/CRC, Boca Raton, FL, 2001.

[2]

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

[3]

T. Bartsch and M. Clapp, Bifurcation theory for symmetric potential operators and the equivariant cup-length, Math. Z., 204 (1990), 341-356.  doi: 10.1007/BF02570878.

[4]

V. Benci and F. Pacella, Morse theory for symmetric functionals on the sphere and an application to a bifurcation problem, Nonlinear Analysis Theory Methods & Applications, 9 (1985), 763-773.  doi: 10.1016/0362-546X(85)90016-1.

[5]

M. S. Berger, Bifurcation theory and the type numbers of Marston Morse, Proc. Nat. Acad. Sci. USA, 69 (1972), 1737-1738.  doi: 10.1073/pnas.69.7.1737.

[6]

M. S. Berger, Nonlinearity and Functional Analysis, Acad. Press, New York-London, 1977.

[7]

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.

[8]

N. A. Bobylev and Y. M. Burman, Morse lemmas for multi-dimensional variational problems, Nonlinear Analysis, 18 (1992), 595-604.  doi: 10.1016/0362-546X(92)90213-X.

[9]

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

[10]

F. E. Browder, Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc., 76 (1970), 999-1005.  doi: 10.1090/S0002-9904-1970-12530-7.

[11]

A. Canino, Variational bifurcation for quasilinear elliptic equations, Calc. Var., 18 (2003), 269-286.  doi: 10.1007/s00526-003-0200-6.

[12]

K.-C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkhäuser, 1993. doi: 10.1007/978-1-4612-0385-8.

[13]

K.-C. Chang, Methods in Nonlinear Analysis, Springer Monogaphs in Mathematics, Springer 2005.

[14]

K.-C. Chang and Z.-Q. Wang, Notes on the bifurcation theorem, J. Fixed Point Theory Appl., 1 (2007), 195-208.  doi: 10.1007/s11784-007-0013-x.

[15]

S.-N. Chow and R. Lauterbach, A bifurcation theorem for critical points of variational problems, Nonlinear Anal., Theory Methods Appl., 12 (1988), 51-61.  doi: 10.1016/0362-546X(88)90012-0.

[16]

S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces, Adv. Nonlinear Stud., 9 (2009), 679-699.  doi: 10.1515/ans-2009-0406.

[17]

D. C. Clark, A variant of the Ljusternik-Schnirelman theory, Indiana Uniw. Math. J., 22 (1972), 65-74.  doi: 10.1512/iumj.1973.22.22008.

[18]

J.-N. Corvellec, Morse theory for continuous functionals, J. Math. Anal. Appl., 196 (1995), 1050–1072. doi: 10.1006/jmaa.1995.1460.

[19]

J.-N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Set-Valued Analysis, 10 (2002), 143-164.  doi: 10.1023/A:1016544301594.

[20]

J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, Mathematical Surveys, No. 11 American Mathematical Society, Providence, R.I. 1964.

[21]

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

[22]

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

[23]

U. Dierkes, S. Hildebrandt and A. J. Tromba, Global Analysis of Minimal Surfaces, Revised and enlarged second edition. Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11706-0.

[24]

S. V. Emelyanov, S. K. Korovin, N. A. Bobylev and A. V. Bulatov, Homotopy of Extremal Problems. Theory and Applications, De Gruyter Series in Nonlinear Analysis and Applications, 11. Walter de Gruyter & Co., Berlin, 2007. doi: 10.1515/9783110893014.

[25]

G. Evéquoz and C. A. Stuart, Hadamard differentiability and bifurcation, Proc. R. Soc. Edinb. A, 137 (2007), 1249-1285.  doi: 10.1017/S0308210506000424.

[26]

E. R. Fadell and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal., 26 (1977), 48-67.  doi: 10.1016/0022-1236(77)90015-5.

[27]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent Math., 45 (1978), 139-174.  doi: 10.1007/BF01390270.

[28]

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

[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.

[30] D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, 2001. 
[31]

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.

[32]

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

[33]

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

[34]

T. Kato, Perturbation Theory for Linear Operators, Second edition., Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976.

[35]

H. Kielhöffer, A bifurcation theorem for potential operators, J. Funct. Anal., 77 (1988), 1-8.  doi: 10.1016/0022-1236(88)90073-0.

[36]

H. Kielhöffer, Bifurcation Theory. An Introduction with Applications to Partial Differential Equations, Second edition. Applied Mathematical Sciences, 156. Springer, New York, 2012. doi: 10.1007/978-1-4614-0502-3.

[37]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, McMillan, New York, 1964.

[38]

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

[39]

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

[40]

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.

[41]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces I, Discrete Contin. Dyn. Syst., $\textsf {33}$ (2013), 2939–2990. doi: 10.3934/dcds.2013.33.2939.

[42]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces II, Topol. Meth. Nonlinear Anal., 44 (2014), 277-335.  doi: 10.12775/TMNA.2014.048.

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[49]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.

[50]

D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology, Second edition. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2012.

[51]

J. B. McLeod and R. E. L. Turner, Bifurcation of nondifferentiable operators with an application to elasticity, Arch. Rational Mech. Anal., 63 (1976-1977), 1-45.  doi: 10.1007/BF00280140.

[52]

D. Motreanu, V. V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[53]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Lecture Notes, 6, Providence, RI : American Mathematical Society, 2001. doi: 10.1090/cln/006.

[54]

R. S. Palais, Ljusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132.  doi: 10.1016/0040-9383(66)90013-9.

[55]

E. Pérez-Chavela, S. Rybicki and D. Strzelecki, Symmetric Liapunov center theorem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 26, 23 pp. doi: 10.1007/s00526-017-1120-1.

[56]

H. Poincare, Oeuvres, Tome VII, (1885), 41–140.

[57]

G. Prodi, Problemi di diramazione per equazioni funzionali. (Italian), Boll. Un. Mat. Ital., 22 (1967), 413-433. 

[58]

P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, in Proc. Sym. on Eigenvalues of Nonlinear Problems (Centro Internaz. Mat. Estivo(C.I.M.E.), III Ciclo, Varenna, 1974), pp. 139-195. Edizioni Cremonese, Rome, 1974.

[59]

P. H. Rabinowitz, A bifurcation theorem for potential operators, J. Funct. Anal., $\textsf{25}$ (1977), 412–424. doi: 10.1016/0022-1236(77)90047-7.

[60]

S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index, Advanced Nonlinear Studies, $\textsf{11}$ (2011), 929–940. doi: 10.1515/ans-2011-0410.

[61]

E. Schmidt, Zur theorie der linearen und nichtlinearen integralgleichungen. III. Teil, Math. Ann., 65 (1908), 370-399.  doi: 10.1007/BF01456418.

[62]

I. V. Skrypnik, Nonlinear Elliptic Equations of a Higher Order, [in Russian], Naukova Dumka, Kiev 1973. 219 pp. doi: 10.1007/BF01097352.

[63]

I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, in: Translations of Mathematical Monographs, vol. 139, Providence, Rhode Island, 1994. doi: 10.1090/mmono/139.

[64]

I. V. Skrypnik, Solvability and properties of solutions of nonlinear elliptic equations, J. Soviet Math., 12 (1979), 555-629.  doi: 10.1007/BF01089138.

[65]

J. Smoller and A. G. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.  doi: 10.1007/BF01231181.

[66]

C. A. Stuart, Bifurcation at isolated singular points of the Hadamard derivative, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1027-1065.  doi: 10.1017/S0308210513000486.

[67]

C. A. Stuart, Bifurcation without Fréchet differentiability at the trivial solution, Math. Methods Appl. Sci., 38 (2015), 3444-3463.  doi: 10.1002/mma.3409.

[68]

C. A. Stuart, Asymptotic bifurcation and second order elliptic equations on $ {\mathbb R}^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1259-1281.  doi: 10.1016/j.anihpc.2014.09.003.

[69]

E. Tonkes, Bifurcation of gradient mappings possessing the Palais-Smale condition, Int. J. Math. Math. Sci., (2011), Art. ID 564930, 14 pp. doi: 10.1155/2011/564930.

[70]

A. J. Tromba, A general approach to Morse theory, J. Differential Geometry, 12 (1977), 47-85.  doi: 10.4310/jdg/1214433845.

[71]

A. J. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem, Math. Ann., 263 (1983), 303-312.  doi: 10.1007/BF01457133.

[72]

K. Uhlenbeck, Morse theory on Banach manifolds, J. Funct. Anal., 10 (1972), 430-445.  doi: 10.1016/0022-1236(72)90039-0.

[73]

Z. Q. Wang, Equivariant Morse theory for isolated critical orbits and its applications to nonlinear problems, Lect. Notes in Math., 1306, Springer, (1988), 202–221. doi: 10.1007/BFb0082935.

[74]

A. G. Wasserman, Equivariant differential topology, Topology, 8 (1969), 127-150.  doi: 10.1016/0040-9383(69)90005-6.

[75]

G. Q. Zhang, A bifurcation theorem, J. Systems Sci. Math. Sci., 4 (1984), 191-195. 

show all references

References:
[1]

A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC Research Notes in Mathematics, 425. Chapman & Hall/CRC, Boca Raton, FL, 2001.

[2]

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

[3]

T. Bartsch and M. Clapp, Bifurcation theory for symmetric potential operators and the equivariant cup-length, Math. Z., 204 (1990), 341-356.  doi: 10.1007/BF02570878.

[4]

V. Benci and F. Pacella, Morse theory for symmetric functionals on the sphere and an application to a bifurcation problem, Nonlinear Analysis Theory Methods & Applications, 9 (1985), 763-773.  doi: 10.1016/0362-546X(85)90016-1.

[5]

M. S. Berger, Bifurcation theory and the type numbers of Marston Morse, Proc. Nat. Acad. Sci. USA, 69 (1972), 1737-1738.  doi: 10.1073/pnas.69.7.1737.

[6]

M. S. Berger, Nonlinearity and Functional Analysis, Acad. Press, New York-London, 1977.

[7]

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.

[8]

N. A. Bobylev and Y. M. Burman, Morse lemmas for multi-dimensional variational problems, Nonlinear Analysis, 18 (1992), 595-604.  doi: 10.1016/0362-546X(92)90213-X.

[9]

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

[10]

F. E. Browder, Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc., 76 (1970), 999-1005.  doi: 10.1090/S0002-9904-1970-12530-7.

[11]

A. Canino, Variational bifurcation for quasilinear elliptic equations, Calc. Var., 18 (2003), 269-286.  doi: 10.1007/s00526-003-0200-6.

[12]

K.-C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkhäuser, 1993. doi: 10.1007/978-1-4612-0385-8.

[13]

K.-C. Chang, Methods in Nonlinear Analysis, Springer Monogaphs in Mathematics, Springer 2005.

[14]

K.-C. Chang and Z.-Q. Wang, Notes on the bifurcation theorem, J. Fixed Point Theory Appl., 1 (2007), 195-208.  doi: 10.1007/s11784-007-0013-x.

[15]

S.-N. Chow and R. Lauterbach, A bifurcation theorem for critical points of variational problems, Nonlinear Anal., Theory Methods Appl., 12 (1988), 51-61.  doi: 10.1016/0362-546X(88)90012-0.

[16]

S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces, Adv. Nonlinear Stud., 9 (2009), 679-699.  doi: 10.1515/ans-2009-0406.

[17]

D. C. Clark, A variant of the Ljusternik-Schnirelman theory, Indiana Uniw. Math. J., 22 (1972), 65-74.  doi: 10.1512/iumj.1973.22.22008.

[18]

J.-N. Corvellec, Morse theory for continuous functionals, J. Math. Anal. Appl., 196 (1995), 1050–1072. doi: 10.1006/jmaa.1995.1460.

[19]

J.-N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Set-Valued Analysis, 10 (2002), 143-164.  doi: 10.1023/A:1016544301594.

[20]

J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, Mathematical Surveys, No. 11 American Mathematical Society, Providence, R.I. 1964.

[21]

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

[22]

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

[23]

U. Dierkes, S. Hildebrandt and A. J. Tromba, Global Analysis of Minimal Surfaces, Revised and enlarged second edition. Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11706-0.

[24]

S. V. Emelyanov, S. K. Korovin, N. A. Bobylev and A. V. Bulatov, Homotopy of Extremal Problems. Theory and Applications, De Gruyter Series in Nonlinear Analysis and Applications, 11. Walter de Gruyter & Co., Berlin, 2007. doi: 10.1515/9783110893014.

[25]

G. Evéquoz and C. A. Stuart, Hadamard differentiability and bifurcation, Proc. R. Soc. Edinb. A, 137 (2007), 1249-1285.  doi: 10.1017/S0308210506000424.

[26]

E. R. Fadell and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal., 26 (1977), 48-67.  doi: 10.1016/0022-1236(77)90015-5.

[27]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent Math., 45 (1978), 139-174.  doi: 10.1007/BF01390270.

[28]

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

[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.

[30] D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, 2001. 
[31]

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.

[32]

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

[33]

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

[34]

T. Kato, Perturbation Theory for Linear Operators, Second edition., Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976.

[35]

H. Kielhöffer, A bifurcation theorem for potential operators, J. Funct. Anal., 77 (1988), 1-8.  doi: 10.1016/0022-1236(88)90073-0.

[36]

H. Kielhöffer, Bifurcation Theory. An Introduction with Applications to Partial Differential Equations, Second edition. Applied Mathematical Sciences, 156. Springer, New York, 2012. doi: 10.1007/978-1-4614-0502-3.

[37]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, McMillan, New York, 1964.

[38]

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

[39]

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

[40]

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.

[41]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces I, Discrete Contin. Dyn. Syst., $\textsf {33}$ (2013), 2939–2990. doi: 10.3934/dcds.2013.33.2939.

[42]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces II, Topol. Meth. Nonlinear Anal., 44 (2014), 277-335.  doi: 10.12775/TMNA.2014.048.

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[49]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.

[50]

D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology, Second edition. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2012.

[51]

J. B. McLeod and R. E. L. Turner, Bifurcation of nondifferentiable operators with an application to elasticity, Arch. Rational Mech. Anal., 63 (1976-1977), 1-45.  doi: 10.1007/BF00280140.

[52]

D. Motreanu, V. V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[53]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Lecture Notes, 6, Providence, RI : American Mathematical Society, 2001. doi: 10.1090/cln/006.

[54]

R. S. Palais, Ljusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132.  doi: 10.1016/0040-9383(66)90013-9.

[55]

E. Pérez-Chavela, S. Rybicki and D. Strzelecki, Symmetric Liapunov center theorem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 26, 23 pp. doi: 10.1007/s00526-017-1120-1.

[56]

H. Poincare, Oeuvres, Tome VII, (1885), 41–140.

[57]

G. Prodi, Problemi di diramazione per equazioni funzionali. (Italian), Boll. Un. Mat. Ital., 22 (1967), 413-433. 

[58]

P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, in Proc. Sym. on Eigenvalues of Nonlinear Problems (Centro Internaz. Mat. Estivo(C.I.M.E.), III Ciclo, Varenna, 1974), pp. 139-195. Edizioni Cremonese, Rome, 1974.

[59]

P. H. Rabinowitz, A bifurcation theorem for potential operators, J. Funct. Anal., $\textsf{25}$ (1977), 412–424. doi: 10.1016/0022-1236(77)90047-7.

[60]

S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index, Advanced Nonlinear Studies, $\textsf{11}$ (2011), 929–940. doi: 10.1515/ans-2011-0410.

[61]

E. Schmidt, Zur theorie der linearen und nichtlinearen integralgleichungen. III. Teil, Math. Ann., 65 (1908), 370-399.  doi: 10.1007/BF01456418.

[62]

I. V. Skrypnik, Nonlinear Elliptic Equations of a Higher Order, [in Russian], Naukova Dumka, Kiev 1973. 219 pp. doi: 10.1007/BF01097352.

[63]

I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, in: Translations of Mathematical Monographs, vol. 139, Providence, Rhode Island, 1994. doi: 10.1090/mmono/139.

[64]

I. V. Skrypnik, Solvability and properties of solutions of nonlinear elliptic equations, J. Soviet Math., 12 (1979), 555-629.  doi: 10.1007/BF01089138.

[65]

J. Smoller and A. G. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.  doi: 10.1007/BF01231181.

[66]

C. A. Stuart, Bifurcation at isolated singular points of the Hadamard derivative, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1027-1065.  doi: 10.1017/S0308210513000486.

[67]

C. A. Stuart, Bifurcation without Fréchet differentiability at the trivial solution, Math. Methods Appl. Sci., 38 (2015), 3444-3463.  doi: 10.1002/mma.3409.

[68]

C. A. Stuart, Asymptotic bifurcation and second order elliptic equations on $ {\mathbb R}^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1259-1281.  doi: 10.1016/j.anihpc.2014.09.003.

[69]

E. Tonkes, Bifurcation of gradient mappings possessing the Palais-Smale condition, Int. J. Math. Math. Sci., (2011), Art. ID 564930, 14 pp. doi: 10.1155/2011/564930.

[70]

A. J. Tromba, A general approach to Morse theory, J. Differential Geometry, 12 (1977), 47-85.  doi: 10.4310/jdg/1214433845.

[71]

A. J. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem, Math. Ann., 263 (1983), 303-312.  doi: 10.1007/BF01457133.

[72]

K. Uhlenbeck, Morse theory on Banach manifolds, J. Funct. Anal., 10 (1972), 430-445.  doi: 10.1016/0022-1236(72)90039-0.

[73]

Z. Q. Wang, Equivariant Morse theory for isolated critical orbits and its applications to nonlinear problems, Lect. Notes in Math., 1306, Springer, (1988), 202–221. doi: 10.1007/BFb0082935.

[74]

A. G. Wasserman, Equivariant differential topology, Topology, 8 (1969), 127-150.  doi: 10.1016/0040-9383(69)90005-6.

[75]

G. Q. Zhang, A bifurcation theorem, J. Systems Sci. Math. Sci., 4 (1984), 191-195. 

[1]

Yunmei Chen, Xianqi Li, Yuyuan Ouyang, Eduardo Pasiliao. Accelerated bregman operator splitting with backtracking. Inverse Problems and Imaging, 2017, 11 (6) : 1047-1070. doi: 10.3934/ipi.2017048

[2]

Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806

[3]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[4]

Lijian Jiang, Craig C. Douglas. Analysis of an operator splitting method in 4D-Var. Conference Publications, 2009, 2009 (Special) : 394-403. doi: 10.3934/proc.2009.2009.394

[5]

Leyu Hu, Wenxing Zhang, Xingju Cai, Deren Han. A parallel operator splitting algorithm for solving constrained total-variation retinex. Inverse Problems and Imaging, 2020, 14 (6) : 1135-1156. doi: 10.3934/ipi.2020058

[6]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[7]

Berat Karaagac. Numerical treatment of Gray-Scott model with operator splitting method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2373-2386. doi: 10.3934/dcdss.2020143

[8]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics and Games, 2021, 8 (4) : 467-486. doi: 10.3934/jdg.2021014

[9]

Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923

[10]

Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835

[11]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

[12]

Lili Yan. Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022034

[13]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215

[14]

Gui-Qiang Chen, Bo Su. A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1587-1606. doi: 10.3934/dcds.2003.9.1587

[15]

Kangkang Deng, Zheng Peng, Jianli Chen. Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1881-1896. doi: 10.3934/jimo.2018127

[16]

Guangcun Lu. Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1317-1368. doi: 10.3934/dcds.2021155

[17]

Kuo-Chih Hung, Shao-Yuan Huang, Shin-Hwa Wang. A global bifurcation theorem for a positone multiparameter problem and its application. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5127-5149. doi: 10.3934/dcds.2017222

[18]

Frank Neubrander, Koray Özer, Lee Windsperger. On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3565-3579. doi: 10.3934/dcdss.2020238

[19]

Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206

[20]

Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (186)
  • HTML views (186)
  • Cited by (0)

Other articles
by authors

[Back to Top]