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

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

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

Christopher Grumiau ^1, , Marco Squassina ^2, and Christophe Troestler ^1,

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

Abstract
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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.

Keywords: mountain pass solutions, Quasi-linear equations, mountain pass algorithm..

Mathematics Subject Classification: Primary: 35J62, 58E05; Secondary: 35D99, 65N12, 35J7.

Citation: Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and 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 $\mathbb{R}$, Disc. Cont. Dyna. Syst. - A, 9 (2003), 55-68. doi: 10.3934/dcds.2003.9.55.
[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.
[3]	M. Caliari and M. Squassina, On a bifurcation value related to quasi-linear Schrödinger equations, J. Fixed Point Theory Appl., to Appear, arXiv:1111.0526v3.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
[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.
[12]	E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in $\mathbb{R}^N2$, J. Math. Anal. Appl., 371 (2010), 465-484. doi: 10.1016/j.jmaa.2010.05.033.
[13]	C. Grumiau and C. Troestler, Convergence of a mountain pass type algorithm for strongly indefinite problems and systems, Preprint, arXiv:1301.1456.
[14]	L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^N2$, Proc. Amer. Math. Soc., 131 (2002), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.
[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.
[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.
[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.
[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.
[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.
[20]	P. Pucci and J. Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and Their Applications, 73, Birkhäuser Verlag, 2007.
[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.
[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.
[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.
[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.

show all references

References:

[1]	A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbb{R}$, Disc. Cont. Dyna. Syst. - A, 9 (2003), 55-68. doi: 10.3934/dcds.2003.9.55.
[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.
[3]	M. Caliari and M. Squassina, On a bifurcation value related to quasi-linear Schrödinger equations, J. Fixed Point Theory Appl., to Appear, arXiv:1111.0526v3.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
[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.
[12]	E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in $\mathbb{R}^N2$, J. Math. Anal. Appl., 371 (2010), 465-484. doi: 10.1016/j.jmaa.2010.05.033.
[13]	C. Grumiau and C. Troestler, Convergence of a mountain pass type algorithm for strongly indefinite problems and systems, Preprint, arXiv:1301.1456.
[14]	L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^N2$, Proc. Amer. Math. Soc., 131 (2002), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.
[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.
[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.
[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.
[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.
[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.
[20]	P. Pucci and J. Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and Their Applications, 73, Birkhäuser Verlag, 2007.
[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.
[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.
[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.
[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.

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