Citation: |
[1] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381.doi: 10.1016/0022-1236(73)90051-7. |
[2] |
J. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984. |
[3] |
F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves, Phys. Rep., 189 (1990), 165-223.doi: 10.1016/0370-1573(90)90093-H. |
[4] |
V. Benci and G. Cerami, Existence of positive solutions of the equation $ - \Delta u + a(x)u = u^{\frac {N + 2} {N - 2}}$ in $\mathbbR^N$, J. Funct. Anal., 88 (1990), 90-117.doi: 10.1016/0022-1236(90)90120-A. |
[5] |
H. Berestycki, T. Gallouët and O. Kavian, Equations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math., 297 (1983), 307-310. |
[6] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.doi: 10.1007/BF00250555. |
[7] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.doi: 10.1007/BF00250556. |
[8] |
J. M. Bezerra do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.doi: 10.1016/j.jde.2009.11.030. |
[9] |
A. Borovskii and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP, 77 (1983), 562-573. |
[10] |
A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schröndinger equation, Commun. Math. Phys., 189 (1997), 73-105.doi: 10.1007/s002200050191. |
[11] |
H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids, B5 (1993), 3539-3550. |
[12] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.doi: 10.1090/S0002-9939-1983-0699419-3. |
[13] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.doi: 10.1002/cpa.3160360405. |
[14] |
J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal., 185 (2007), 185-200.doi: 10.1007/s00205-006-0019-3. |
[15] |
J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differential Equations, 18 (2003), 207-219.doi: 10.1007/s00526-002-0191-8. |
[16] |
K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.doi: 10.1007/978-1-4612-0385-8. |
[17] |
S. Cingolani and N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13. |
[18] |
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. |
[19] |
X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett., 70 (1993), 2082-2085. |
[20] |
G. M. Figueiredo, N. Ikoma and J. R. Santos Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979.doi: 10.1007/s00205-014-0747-8. |
[21] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.doi: 10.1016/0022-1236(86)90096-0. |
[22] |
C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Commun. Partial Differential Equations, 21 (1996), 787-820.doi: 10.1080/03605309608821208. |
[23] |
J. Garcia Azorero and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a non-symmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.doi: 10.1090/S0002-9947-1991-1083144-2. |
[24] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983.doi: 10.1007/978-3-642-61798-0. |
[25] |
J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. |
[26] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys., 37 (1980), 83-87.doi: 10.1007/BF01325508. |
[27] |
Y. He and G. Li, The existence and concentration of weak solutions to a class of $p$-Laplacian type problems in unbounded domains, Sci. China Math., 57 (2014), 1927-1952.doi: 10.1007/s11425-014-4830-2. |
[28] |
L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.doi: 10.1090/S0002-9939-02-06821-1. |
[29] |
L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbbR^N$, Indiana Univ. Math. J., 54 (2005), 443-464.doi: 10.1512/iumj.2005.54.2502. |
[30] |
S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3262-3267.doi: 10.1143/JPSJ.50.3262. |
[31] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons in superfluid films, Phys. Rep., 194 (1990), 117-238. |
[32] |
G. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27-36.doi: 10.5186/aasfm.1990.1521. |
[33] |
G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbbR^N$, Commun. Partial Differential Equations, 14 (1989), 1291-1314.doi: 10.1080/03605308908820654. |
[34] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 1 (1984), 223-283. |
[35] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1, Rev. Mat. H. Iberoamericano, 1 (1985), 145-201.doi: 10.4171/RMI/6. |
[36] |
X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.doi: 10.1090/S0002-9939-2012-11293-6. |
[37] |
X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.doi: 10.1016/j.jde.2012.09.006. |
[38] |
E. Laedke and K. Spatschek, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.doi: 10.1063/1.525675. |
[39] |
J. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2003), 441-448.doi: 10.1090/S0002-9939-02-06783-7. |
[40] |
J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differential Equations, 29 (2004), 879-901.doi: 10.1081/PDE-120037335. |
[41] |
J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493.doi: 10.1016/S0022-0396(02)00064-5. |
[42] |
J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations, 38 (2010), 275-315.doi: 10.1007/s00526-009-0286-6. |
[43] |
V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.doi: 10.1016/0370-1573(84)90106-6. |
[44] |
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical growth in $\mathbbR^N$, J. Differential Equations, 229 (2006), 570-587.doi: 10.1016/j.jde.2006.07.001. |
[45] |
O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbbR^N$ with critical growth, Nonlinear Anal., 29 (1997), 773-781.doi: 10.1016/S0362-546X(96)00087-9. |
[46] |
E. S. Noussair, C. A. Swanson and J. F. Yang, Quasilinear elliptic problems with critical exponents, Nonlinear Anal., 20 (1993), 285-301.doi: 10.1016/0362-546X(93)90164-N. |
[47] |
W. M. Ni and I. Takagi, On the shape of least energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.doi: 10.1002/cpa.3160440705. |
[48] |
W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.doi: 10.1215/S0012-7094-93-07004-4. |
[49] |
W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.doi: 10.1002/cpa.3160480704. |
[50] |
Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class $( V )_a$, Commun. Partial Differential Equations, 13 (1988), 1499-1519.doi: 10.1080/03605308808820585. |
[51] |
Y. G. Oh, Corrections to existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class $( V )_a$, Commun. Partial Differential Equations, 14 (1989), 833-834. |
[52] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys., 131 (1990), 223-253.doi: 10.1007/BF02161413. |
[53] |
M. del Pino and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.doi: 10.1007/BF01189950. |
[54] |
M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.doi: 10.1006/jfan.1996.3085. |
[55] |
M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.doi: 10.1007/s002080200327. |
[56] |
M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, 15 (1998), 127-149.doi: 10.1016/S0294-1449(97)89296-7. |
[57] |
M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.doi: 10.1007/s005260100105. |
[58] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.doi: 10.1512/iumj.1986.35.35036. |
[59] |
G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A., 110 (1982), 41-80.doi: 10.1016/0378-4371(82)90104-2. |
[60] |
P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.doi: 10.1007/BF00946631. |
[61] |
M. Ramos, Z. Q. Wang and M. Willem, Positive solutions for elliptic equations with critical growth in unbounded domains, Calculus of Variations and Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 410 (2000), 192-199. |
[62] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev., 50 (1994), 687-689.doi: 10.1103/PhysRevE.50.R687. |
[63] |
S. Takeno and S. Homma, Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitation, Progr. Theoret. Physics, 65 (1981), 1844-1857.doi: 10.1143/PTP.65.1844. |
[64] |
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.doi: 10.1016/0022-0396(84)90105-0. |
[65] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244.doi: 10.1007/BF02096642. |
[66] |
Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations, Nonlinear Differ. Equ. Appl., 19 (2012), 19-47.doi: 10.1007/s00030-011-0116-3. |
[67] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.doi: 10.1007/978-1-4612-4146-1. |
[68] |
J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, J. Lond. Math. Soc., 90 (2014), 827-844.doi: 10.1112/jlms/jdu054. |
[69] |
X. Zhu and J. Yang, Regularity for quasilinear elliptic equations in involving critical Sobolev exponent, System Sci. Math., 9 (1989), 47-52. |