A Ljusternik-Schnirelman minimax algorithm for finding equality constrained saddle points and its application for solving eigen problems: part I. Algorithm and global convergence

1. Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some ellptic problems. J. Funct. Anal. 122, 519–543 (1994)

   Article  MathSciNet  MATH  Google Scholar 

2. Bao, W., Du, Q.: Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25, 1674–1697 (2004)

   Article  MathSciNet  MATH  Google Scholar 

3. Bartsch, T., Willem, M.: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123, 3555–3561 (1995)

   Article  MathSciNet  MATH  Google Scholar 

4. Chang, S., Lin, C., Lin, T., Lin, W.: Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Physica D 196, 341–361 (2004)

   Article  MathSciNet  MATH  Google Scholar 

5. Chen, X., Zhou, J.: A local min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems. Math. Comput. 79, 2213–2236 (2010)

   Article  MathSciNet  MATH  Google Scholar 

6. Chen, X., Zhou, J., Yao, X.: A numerical method for finding multiple co-existing solutions to nonlinear cooperative systems. Appl. Numer. Math. 58, 1614–1627 (2008)

   Article  MathSciNet  MATH  Google Scholar 

7. Choi, Y.S., McKenna, P.J.: A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Anal. 20, 417–437 (1993)

   Article  MathSciNet  MATH  Google Scholar 

8. Ding, Z., Costa, D., Chen, G.: A high linking method for sign changing solutions for semilinear elliptic equations. Nonlinear Anal. 38, 151–172 (1999)

   Article  MathSciNet  MATH  Google Scholar 

9. Perez-Garcia, V.M., Michinel, H., Cirac, J.I., Lewenstein, M., Zoller, P.: Low energy excitations of a Bose-Einstein condensate: A time-dependent variational analysis. Phys. Rev. Lett. 77, 5320 (1996)

   Article  Google Scholar 

10. Garcia-Ripoll, J.J., Konotop, V.V., Malomed, B.M., Perez-Garcia, V.M.: A quasilocal Gross-Pitaevskii equation for attractive Bose-Einstein condensate. Math. Comput. Simul. 62, 21–30 (2003)

    Article  MATH  Google Scholar 

11. Gross, E.P.: Structure of a quantized vortex in boson systems. Nuovo Cimento 20, 454–477 (1961)

    Article  MathSciNet  MATH  Google Scholar 

12. Li, Y., Zhou, J.: A minimax method for finding multiple critical points and its applications to nonlinear PDEs. SIAM J. Sci. Comput. 23, 840–865 (2001)

    Article  MathSciNet  MATH  Google Scholar 

13. Li, Y., Zhou, J.: Convergence results of a local minimax method for finding multiple critical points. SIAM J. Sci. Comput. 24, 865–885 (2002)

    Article  MathSciNet  MATH  Google Scholar 

14. Milner, J.: Morse Theory. Princeton University Press (1963)

15. Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Soviet Phys. JETP 13, 451–454 (1961)

    MathSciNet  Google Scholar 

16. Yao, X.: A minimax method for finding saddle critical points of upper semi-differentiable locally Lipschitz continuous functional in Hilbert space and its convergence. Math. Comput. 82, 2087–2136 (2013)

    Article  MathSciNet  MATH  Google Scholar 

17. Yao, X.: Convergence analysis of a minimax method for finding multiple solutions of semilinear elliptic equation: Part I-On polyhedral domain. J. Sci. Comput. 62, 652–673 (2015)

    Article  MathSciNet  MATH  Google Scholar 

18. Yao, X.: A minimax method for finding saddle points of upper semi-differentiable locally Lipschitz continuous functional in Banach space and its convergence. J. Comput. Appl. Math. 296, 528–549 (2016)

    Article  MathSciNet  MATH  Google Scholar 

19. Yao, X.: Convergence analysis of a minimax method for finding multiple solutions of hemivariational inequality in Hilbert space. Adv. Comput. Math. 42, 1331–1362 (2016)

    Article  MathSciNet  MATH  Google Scholar 

20. Yao, X.: Ljusternik-Schnirelman minimax algorithms and an application for finding multiple negative energy solutions of semilinear elliptic Dirichlet problem involving concave and convex nonlinearities Part I. Algorithms and convergence. J. Sci. Comput. 66, 19–40 (2016)

    Article  MathSciNet  MATH  Google Scholar 

21. Yao, X.: Two classes of Ljusternik-Schnirelman minimax algorithms and an application for finding multiple negative energy solutions of a class of p-Laplacian equations. J. Comput. Appl. Math. 342, 495–520 (2018)

    Article  MathSciNet  MATH  Google Scholar 

22. Yao, X., Zhou, J.: A local minimax characterization for computing multiple nonsmooth saddle critical points. Math. Program. 104(2–3), Ser. B, 749–760 (2005)

    Article  MathSciNet  MATH  Google Scholar 

23. Yao, X., Zhou, J.: A minimax method for finding multiple critical points in Banach spaces and its application to quasi-linear elliptic PDE. SIAM J. Sci. Comput. 26, 1796–1809 (2005)

    Article  MathSciNet  MATH  Google Scholar 

24. Yao, X., Zhou, J.: Unified convergence results on a minimax algorithm for finding multiple critical points in Banach spaces. SIAM J. Num. Anal. 45, 1330–1347 (2007)

    Article  MathSciNet  MATH  Google Scholar 

25. Yao, X., Zhou, J.: Numerical methods for computing nonlinear eigenpairs: Part I. Isohomogeneous cases. SIAM J. Sci. Comput. 29, 1355–1374 (2007)

    Article  MathSciNet  MATH  Google Scholar 

26. Yao, X., Zhou, J.: Numerical methods for computing nonlinear eigenpairs: Part II. Non-Isohomogeneous cases. SIAM J. Sci. Comput. 30, 937–956 (2008)

    Article  MathSciNet  MATH  Google Scholar 

27. Zeidler, E.: Nonlinear Functional Analysis and its Applications III. Springer, New York (1985)

    Book  MATH  Google Scholar 

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