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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 |
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. |
[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., ().
|
[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 $\mathbbR^N$, 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, ().
|
[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. |
[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 $\mathbbR$, 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., ().
|
[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 $\mathbbR^N$, 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, ().
|
[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. |
[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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