March  2022, 42(3): 1317-1368. doi: 10.3934/dcds.2021155

Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems

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 second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear elliptic boundary value problems of higher order.

Citation: 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
References:
[1]

R. Adams and J. J. F. Fournier, Sobolev Spaces, Second Edition, Pure and Applied Mathematics Series, Vol. 140, Academic Press, 2003.

[2]

E. Benincasa and A. Canino, A bifurcation result of Böhme-Marino type for quasilinear elliptic equations, Topol. Meth. Nonlinear Anal., 31 (2008), 1-17. 

[3]

R. G. Bettiol and P. Piccione, Delaunay-type hypersurfaces in cohomogeneity one manifolds, International Mathematics Research Notices, 2016 (2016), 3124-3162.  doi: 10.1093/imrn/rnv231.

[4]

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.

[5]

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

[6]

A. Y. Borisovich, Functional-topological properties of the Plateau operator and applications to the study of bifurcations in problems of geometry and hydrodynamics, Minimal Surfaces, 287–330, Adv. Soviet Math., 15, Amer. Math. Soc., Providence, RI, 1993.

[7]

A. Y. Borisovich and W. Marzantowicz, Bifurcation of the equivariant minimal interfaces in a hydromechanics problem, Abstr. Appl. Anal., 1 (1996), 291-304.  doi: 10.1155/S1085337596000152.

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[9]

F. E. Browder, Nonlinear elliptic boundary value problems II, Trans. Amer. Math. Soc., 117 (1965), 530-550.  doi: 10.1090/S0002-9947-1965-0173846-9.

[10]

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

[11]

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

[12]

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

[13]

G. CoxC. K. R. T. Jones and J. L. Marzuola, A Morse index theorem for elliptic operators on bounded domains, Comm. Partial Differential Equations, 40 (2015), 1467-1497.  doi: 10.1080/03605302.2015.1025979.

[14]

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

[15]

H. I. Elĭasson, Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194.  doi: 10.4310/jdg/1214427887.

[16]

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.

[17]

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

[18]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies, Princeton Univ. Press, 1983.

[19]

H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, In Mem. Amer. Math. Soc., 226 2013, vi+60 pp. doi: 10.1090/S0065-9266-2013-00676-4.

[20]

M. C. Irwin, On the smoothness of the composition map, Quart. J. Math. Oxford (2), 23 (1972), 113-133.  doi: 10.1093/qmath/23.2.113.

[21]

G. Lu, Morse theory methods for quasi-linear elliptic systems of higher order, arXiv: 1702.06667.

[22]

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.

[23]

G. Lu, Parameterized splitting theorems and bifurcations for potential operators, Part I, Discrete Contin. Dyn. Syst., (2021) doi: 10.3934/dcds.2021154.

[24]

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

[25]

G. Lu, Bifurcations for solutions of Hamiltonian and Lagrangian systems, In Progress.

[26]

G. Lu, Bifurcations via saddle point reduction methods, In Progress.

[27]

G. Lu, Bifurcation aspects for geometrical variational problems, In preparation.

[28]

T.-W. Ma, Higher chain formula proved by combinatorics, Electronic Journal of Combinatorics, 16 (2009), Note 21, 7 pp. doi: 10.37236/259.

[29]

C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Reprint of the 1966 Classics in Mathematics. Springer-Verlag, Berlin, 200C. doi: 10.1007/978-3-540-69952-1.

[30]

R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, 1968, 44.

[31]

P. Piccione and D. V. Tausk, On the Banach differential structure for sets of maps on non-compact domains, Nonlinear Analysis, 46 (2001), 245-265.  doi: 10.1016/S0362-546X(00)00116-4.

[32]

A. Portaluri and N. Waterstraat, On bifurcation for semilinear elliptic Dirichlet problems and the Morse-Smale index theorem, J. Math. Anal. Appl., 408 (2013), 572-575.  doi: 10.1016/j.jmaa.2013.06.037.

[33]

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

[34]

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

[35]

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.

[36]

S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. Math., 80 (1964), 382-396.  doi: 10.2307/1970398.

[37]

S. Smale, On the Morse index theorem, J. Math. Mech., 14 (1965), 1049-1055.  doi: 10.1111/j.1467-9876.1965.tb00656.x.

[38]

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.

[39]

R. C. Swanson, Fredholm intersection theory and elliptic boundary deformation problems, II, J. Diff. Equa., 28 (1978), 202-219.  doi: 10.1016/0022-0396(78)90067-0.

[40]

K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geometry, 8 (1973), 555-564.  doi: 10.4310/jdg/1214431958.

[41]

V. Volpert, Elliptic Partial Differential Equations. Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains, Monographs in Mathematics, 101. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0346-0537-3.

[42]

C. Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry, math.SG, arXiv: 1011.1690V2.

show all references

References:
[1]

R. Adams and J. J. F. Fournier, Sobolev Spaces, Second Edition, Pure and Applied Mathematics Series, Vol. 140, Academic Press, 2003.

[2]

E. Benincasa and A. Canino, A bifurcation result of Böhme-Marino type for quasilinear elliptic equations, Topol. Meth. Nonlinear Anal., 31 (2008), 1-17. 

[3]

R. G. Bettiol and P. Piccione, Delaunay-type hypersurfaces in cohomogeneity one manifolds, International Mathematics Research Notices, 2016 (2016), 3124-3162.  doi: 10.1093/imrn/rnv231.

[4]

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.

[5]

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

[6]

A. Y. Borisovich, Functional-topological properties of the Plateau operator and applications to the study of bifurcations in problems of geometry and hydrodynamics, Minimal Surfaces, 287–330, Adv. Soviet Math., 15, Amer. Math. Soc., Providence, RI, 1993.

[7]

A. Y. Borisovich and W. Marzantowicz, Bifurcation of the equivariant minimal interfaces in a hydromechanics problem, Abstr. Appl. Anal., 1 (1996), 291-304.  doi: 10.1155/S1085337596000152.

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[9]

F. E. Browder, Nonlinear elliptic boundary value problems II, Trans. Amer. Math. Soc., 117 (1965), 530-550.  doi: 10.1090/S0002-9947-1965-0173846-9.

[10]

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

[11]

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

[12]

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

[13]

G. CoxC. K. R. T. Jones and J. L. Marzuola, A Morse index theorem for elliptic operators on bounded domains, Comm. Partial Differential Equations, 40 (2015), 1467-1497.  doi: 10.1080/03605302.2015.1025979.

[14]

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

[15]

H. I. Elĭasson, Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194.  doi: 10.4310/jdg/1214427887.

[16]

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.

[17]

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

[18]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies, Princeton Univ. Press, 1983.

[19]

H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, In Mem. Amer. Math. Soc., 226 2013, vi+60 pp. doi: 10.1090/S0065-9266-2013-00676-4.

[20]

M. C. Irwin, On the smoothness of the composition map, Quart. J. Math. Oxford (2), 23 (1972), 113-133.  doi: 10.1093/qmath/23.2.113.

[21]

G. Lu, Morse theory methods for quasi-linear elliptic systems of higher order, arXiv: 1702.06667.

[22]

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.

[23]

G. Lu, Parameterized splitting theorems and bifurcations for potential operators, Part I, Discrete Contin. Dyn. Syst., (2021) doi: 10.3934/dcds.2021154.

[24]

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

[25]

G. Lu, Bifurcations for solutions of Hamiltonian and Lagrangian systems, In Progress.

[26]

G. Lu, Bifurcations via saddle point reduction methods, In Progress.

[27]

G. Lu, Bifurcation aspects for geometrical variational problems, In preparation.

[28]

T.-W. Ma, Higher chain formula proved by combinatorics, Electronic Journal of Combinatorics, 16 (2009), Note 21, 7 pp. doi: 10.37236/259.

[29]

C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Reprint of the 1966 Classics in Mathematics. Springer-Verlag, Berlin, 200C. doi: 10.1007/978-3-540-69952-1.

[30]

R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, 1968, 44.

[31]

P. Piccione and D. V. Tausk, On the Banach differential structure for sets of maps on non-compact domains, Nonlinear Analysis, 46 (2001), 245-265.  doi: 10.1016/S0362-546X(00)00116-4.

[32]

A. Portaluri and N. Waterstraat, On bifurcation for semilinear elliptic Dirichlet problems and the Morse-Smale index theorem, J. Math. Anal. Appl., 408 (2013), 572-575.  doi: 10.1016/j.jmaa.2013.06.037.

[33]

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

[34]

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

[35]

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.

[36]

S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. Math., 80 (1964), 382-396.  doi: 10.2307/1970398.

[37]

S. Smale, On the Morse index theorem, J. Math. Mech., 14 (1965), 1049-1055.  doi: 10.1111/j.1467-9876.1965.tb00656.x.

[38]

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.

[39]

R. C. Swanson, Fredholm intersection theory and elliptic boundary deformation problems, II, J. Diff. Equa., 28 (1978), 202-219.  doi: 10.1016/0022-0396(78)90067-0.

[40]

K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geometry, 8 (1973), 555-564.  doi: 10.4310/jdg/1214431958.

[41]

V. Volpert, Elliptic Partial Differential Equations. Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains, Monographs in Mathematics, 101. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0346-0537-3.

[42]

C. Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry, math.SG, arXiv: 1011.1690V2.

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