doi: 10.3934/dcds.2021155
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[42]

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

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[42]

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

[1]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[2]

Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020

[3]

Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029

[4]

Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082

[5]

Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020100

[6]

Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635

[7]

Guangying Lv, Mingxin Wang. Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 415-433. doi: 10.3934/dcdsb.2010.13.415

[8]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

[9]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899

[10]

Boris Buffoni, Laurent Landry. Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 75-116. doi: 10.3934/dcds.2010.27.75

[11]

Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009

[12]

Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026

[13]

Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268

[14]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[15]

Osama Moaaz, Omar Bazighifan. Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2465-2473. doi: 10.3934/dcdss.2020136

[16]

Massimo Lanza de Cristoforis, aolo Musolino. A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2509-2542. doi: 10.3934/cpaa.2014.13.2509

[17]

Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501

[18]

Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184

[19]

Simone Creo, Valerio Regis Durante. Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 65-90. doi: 10.3934/dcdss.2019005

[20]

Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113

2020 Impact Factor: 1.392

Article outline

Figures and Tables

[Back to Top]