Quasilinear elliptic equations with critical growth via perturbation method

Journal of Differential Equations - Tập 254 - Trang 102-124 - 2013
Xiang-Qing Liu1, Jia-Quan Liu2, Zhi-Qiang Wang3,4
1Department of Mathematics, Yunnan Normal University, Kunming 650092, PR China
2LMAM, School of Mathematical Science, Peking University, Beijing 100871, PR China
3Chern Institute of Mathematics, Nankai University, Tianjin 300071, PR China
4Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

Tài liệu tham khảo

Ambrosetti, 2003, Positive solutions to a class of quasilinear elliptic equations on R, Discrete Contin. Dyn. Syst., 9, 55, 10.3934/dcds.2003.9.55 Arcoya, 1999, Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl., 6, 79, 10.1007/s000300050066 Alves, 2009, Multiplicity of positive solutions for a class of quasilinear problems, Adv. Differential Equations, 14, 911, 10.57262/ade/1355863334 Brezis, 1983, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437, 10.1002/cpa.3160360405 Brihaye, 2006, Solitons on nanotubes and fullerenes as solutions of a modified nonlinear Schrödinger equation, 135 Brizhik, 2003, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16, 1481, 10.1088/0951-7715/16/4/317 Brüll, 1986, Solitary waves for quasilinear Schrödinger equations, Expo. Math., 4, 278 Canino, 1995, Nonsmooth critical point theory and quasilinear elliptic equations, vol. 472, 1 Colin, 2002, On the local well-posedness of quasilinear Schrödinger equations in arbitrary space dimension, Comm. Partial Differential Equations, 27, 325, 10.1081/PDE-120002789 Colin, 2004, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56, 213, 10.1016/j.na.2003.09.008 Colin, 2010, Stability and instability results for standing waves of quasilinear Schrödinger equations, Nonlinearity, 23, 1353, 10.1088/0951-7715/23/6/006 de Bouard, 1997, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189, 73, 10.1007/s002200050191 do Ó, 2010, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248, 722, 10.1016/j.jde.2009.11.030 do Ó, 2010, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations, 38, 275, 10.1007/s00526-009-0286-6 Han, 2000 Hartmann, 2003, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B, 68, 184302, 10.1103/PhysRevB.68.184302 Hasse, 1980, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37, 83, 10.1007/BF01325508 Kenig, 2004, The Cauchy problem for quasilinear Schrödinger equations, Invent. Math., 158, 343, 10.1007/s00222-004-0373-4 Kurihura, 1981, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50, 3262, 10.1143/JPSJ.50.3262 Lange, 1999, Nash–Moser methods for the solutions of quasilinear Schrödinger equations, Comm. Partial Differential Equations, 24, 1399, 10.1080/03605309908821469 Lins, 2009, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71, 2890, 10.1016/j.na.2009.01.171 Lions, 1984, The concentration–compactness principle in the calculus of variations. The locally compact case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 109, 10.1016/S0294-1449(16)30428-0 Liu, 2003, Soliton solutions for quasilinear Schrödinger equations, II, J. Differential Equations, 187, 473, 10.1016/S0022-0396(02)00064-5 Liu, 2004, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29, 879, 10.1081/PDE-120037335 Liu, 2003, Soliton solutions for quasilinear Schrödinger equations, I, Proc. Amer. Math. Soc., 131, 441, 10.1090/S0002-9939-02-06783-7 X.-Q. Liu, J.-Q. Liu, Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., in press, http://dx.doi.org/10.1090/S0002-9939(2012)11293-6, electronically online on May 2012. X.-Q. Liu, J.-Q. Liu, Z.-Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, in press, http://dx.doi.org/10.1007/s00526-012-0497-0, electronically online on January 2012. Moameni, 2006, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical growth in R, J. Differential Equations, 229, 570, 10.1016/j.jde.2006.07.001 Moameni, 2007, On a class of periodic quasilinear Schrödinger equations involving critical growth in R2, J. Math. Anal. Appl., 334, 775, 10.1016/j.jmaa.2007.01.020 Poppenberg, 2001, On the local well posedness of quasilinear Schrödinger equations in arbitrary space dimension, J. Differential Equations, 172, 83, 10.1006/jdeq.2000.3853 Poppenberg, 2002, On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14, 329, 10.1007/s005260100105 Pucci, 1986, A general variational identity, Indiana Univ. Math. J., 35, 681, 10.1512/iumj.1986.35.35036 Rabinowitz, 1986, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 Silva, 2010, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39, 1, 10.1007/s00526-009-0299-1 Silva, 2010, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72, 2935, 10.1016/j.na.2009.11.037