American Institute of Mathematical Sciences

July  2013, 18(5): 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation

 1 Université de Mons, Institut Complexys, Département de Mathématique, Service d'Analyse Numérique, Place du Parc, 20, B-7000 Mons, Belgium, Belgium 2 Dipartimento di Informatica, Università degli Studi di Verona, Cá Vignal 2, Strada Le Grazie 15, I-37134 Veron

Received  May 2012 Revised  December 2012 Published  March 2013

We discuss the application of the Mountain Pass Algorithm to the so-called quasi-linear Schrödinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm cannot directly be used. A change of variable allows us to deal with the lack of regularity. We establish the convergence of a mountain pass algorithm in this setting. Some numerical experiments are also performed and lead to some conjectures.
Citation: 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
References:
 [1] A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbbR$, Disc. Cont. Dyna. Syst. - A, 9 (2003), 55-68. doi: 10.3934/dcds.2003.9.55.  Google Scholar [2] M. Caliari and M. Squassina, Numerical computation of soliton dynamics for NLS equations in a driving potential, Electron. J. Differential Equations, 89 (2010), 1-12, arXiv:0908.3648.  Google Scholar [3] M. Caliari and M. Squassina, On a bifurcation value related to quasi-linear Schrödinger equations,, J. Fixed Point Theory Appl., ().   Google Scholar [4] Y. S. Choi and P. J. McKenna, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Anal., 20 (1993), 417-437. doi: 10.1016/0362-546X(93)90147-K.  Google Scholar [5] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar [6] M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.  Google Scholar [7] J.-N. Corvellec, M. Degiovanni and M. Marzocchi, Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal., 1 (1993), 151-171.  Google Scholar [8] W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.  Google Scholar [9] J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calculus of Variations, 38 (2010), 275-315. doi: 10.1007/s00526-009-0286-6.  Google Scholar [10] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar [11] F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179, arXiv:1108.0207. doi: 10.1515/ana-2011-0001.  Google Scholar [12] E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in $\mathbbR^N$, J. Math. Anal. Appl., 371 (2010), 465-484. doi: 10.1016/j.jmaa.2010.05.033.  Google Scholar [13] C. Grumiau and C. Troestler, Convergence of a mountain pass type algorithm for strongly indefinite problems and systems,, Preprint, ().   Google Scholar [14] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Math. Soc., 131 (2002), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar [15] A. S. Lewis and C. H. J. Pang, Level set methods for finding critical points of mountain pass type, Nonlinear Analysis, 74 (2011), 4058-4082. doi: 10.1016/j.na.2011.03.039.  Google Scholar [16] Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear elliptic pde's, SIAM Sci. Comp., 23 (2001), 840-865. doi: 10.1137/S1064827599365641.  Google Scholar [17] Y. Li and J. Zhou, Convergence results of a local minimax method for finding multiple critical points, SIAM Sci. Comp., 24 (2002), 865-885. doi: 10.1137/S1064827500379732.  Google Scholar [18] E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448. doi: 10.1007/BF01394245.  Google Scholar [19] J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasi-linear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar [20] P. Pucci and J. Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and Their Applications, 73, Birkhäuser Verlag, 2007.  Google Scholar [21] J. R. Shewchuk, Delaunay refinement algorithms for triangular mesh generation, Computational Geometry: Theory and Applications, 22 (2002), 21-74. doi: 10.1016/S0925-7721(01)00047-5.  Google Scholar [22] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar [23] N. Tacheny and C. Troestler, A mountain pass algorithm with projector, J. Comput. Appl. Math., 236 (2012), 2025-2036. doi: 10.1016/j.cam.2011.11.011.  Google Scholar [24] M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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References:
 [1] A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbbR$, Disc. Cont. Dyna. Syst. - A, 9 (2003), 55-68. doi: 10.3934/dcds.2003.9.55.  Google Scholar [2] M. Caliari and M. Squassina, Numerical computation of soliton dynamics for NLS equations in a driving potential, Electron. J. Differential Equations, 89 (2010), 1-12, arXiv:0908.3648.  Google Scholar [3] M. Caliari and M. Squassina, On a bifurcation value related to quasi-linear Schrödinger equations,, J. Fixed Point Theory Appl., ().   Google Scholar [4] Y. S. Choi and P. J. McKenna, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Anal., 20 (1993), 417-437. doi: 10.1016/0362-546X(93)90147-K.  Google Scholar [5] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar [6] M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.  Google Scholar [7] J.-N. Corvellec, M. Degiovanni and M. Marzocchi, Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal., 1 (1993), 151-171.  Google Scholar [8] W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.  Google Scholar [9] J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calculus of Variations, 38 (2010), 275-315. doi: 10.1007/s00526-009-0286-6.  Google Scholar [10] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar [11] F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179, arXiv:1108.0207. doi: 10.1515/ana-2011-0001.  Google Scholar [12] E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in $\mathbbR^N$, J. Math. Anal. Appl., 371 (2010), 465-484. doi: 10.1016/j.jmaa.2010.05.033.  Google Scholar [13] C. Grumiau and C. Troestler, Convergence of a mountain pass type algorithm for strongly indefinite problems and systems,, Preprint, ().   Google Scholar [14] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Math. Soc., 131 (2002), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar [15] A. S. Lewis and C. H. J. Pang, Level set methods for finding critical points of mountain pass type, Nonlinear Analysis, 74 (2011), 4058-4082. doi: 10.1016/j.na.2011.03.039.  Google Scholar [16] Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear elliptic pde's, SIAM Sci. Comp., 23 (2001), 840-865. doi: 10.1137/S1064827599365641.  Google Scholar [17] Y. Li and J. Zhou, Convergence results of a local minimax method for finding multiple critical points, SIAM Sci. Comp., 24 (2002), 865-885. doi: 10.1137/S1064827500379732.  Google Scholar [18] E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448. doi: 10.1007/BF01394245.  Google Scholar [19] J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasi-linear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar [20] P. Pucci and J. Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and Their Applications, 73, Birkhäuser Verlag, 2007.  Google Scholar [21] J. R. Shewchuk, Delaunay refinement algorithms for triangular mesh generation, Computational Geometry: Theory and Applications, 22 (2002), 21-74. doi: 10.1016/S0925-7721(01)00047-5.  Google Scholar [22] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar [23] N. Tacheny and C. Troestler, A mountain pass algorithm with projector, J. Comput. Appl. Math., 236 (2012), 2025-2036. doi: 10.1016/j.cam.2011.11.011.  Google Scholar [24] M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar
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