-
Previous Article
Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions
- MCRF Home
- This Issue
-
Next Article
An optimal control model of carbon reduction and trading
Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity
1. | School of Mathematics and Statistics, South-Central University For Nationalities, Wuhan, 430074 |
2. | Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079 |
References:
[1] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications,, J. Funct. Anal., 14 (1973), 349.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
J. Aubin and I. Ekeland, Applied Nonlinear Analysis,, John Wiley & Sons, (1984).
|
[3] |
F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves,, Phys. Rep., 189 (1990), 165.
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.
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.
|
[6] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.
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.
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.
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. Google Scholar |
[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.
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. Google Scholar |
[12] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.
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.
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.
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.
doi: 10.1007/s00526-002-0191-8. |
[16] |
K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (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.
|
[18] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, Nonlinear Anal., 56 (2004), 213.
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. Google Scholar |
[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.
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.
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.
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.
doi: 10.1090/S0002-9947-1991-1083144-2. |
[24] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (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.
|
[26] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, Z. Phys., 37 (1980), 83.
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.
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.
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.
doi: 10.1512/iumj.2005.54.2502. |
[30] |
S. Kurihura, Large-amplitude quasi-solitons in superfluids films,, J. Phys. Soc. Jpn., 50 (1981), 3262.
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. Google Scholar |
[32] |
G. Li, Some properties of weak solutions of nonlinear scalar field equations,, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27.
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.
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.
|
[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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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. Google Scholar |
[52] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Commun. Math. Phys., 131 (1990), 223.
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.
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.
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.
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.
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.
doi: 10.1007/s005260100105. |
[58] |
P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681.
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.
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.
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, 410 (2000), 192.
|
[62] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, Phys. Rev., 50 (1994), 687.
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.
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.
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.
doi: 10.1007/BF02096642. |
[66] |
Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations,, Nonlinear Differ. Equ. Appl., 19 (2012), 19.
doi: 10.1007/s00030-011-0116-3. |
[67] |
M. Willem, Minimax Theorems,, Birkhäuser, (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.
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.
|
show all references
References:
[1] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications,, J. Funct. Anal., 14 (1973), 349.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
J. Aubin and I. Ekeland, Applied Nonlinear Analysis,, John Wiley & Sons, (1984).
|
[3] |
F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves,, Phys. Rep., 189 (1990), 165.
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.
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.
|
[6] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.
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.
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.
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. Google Scholar |
[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.
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. Google Scholar |
[12] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.
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.
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.
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.
doi: 10.1007/s00526-002-0191-8. |
[16] |
K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (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.
|
[18] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, Nonlinear Anal., 56 (2004), 213.
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. Google Scholar |
[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.
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.
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.
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.
doi: 10.1090/S0002-9947-1991-1083144-2. |
[24] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (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.
|
[26] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, Z. Phys., 37 (1980), 83.
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.
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.
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.
doi: 10.1512/iumj.2005.54.2502. |
[30] |
S. Kurihura, Large-amplitude quasi-solitons in superfluids films,, J. Phys. Soc. Jpn., 50 (1981), 3262.
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. Google Scholar |
[32] |
G. Li, Some properties of weak solutions of nonlinear scalar field equations,, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27.
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.
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.
|
[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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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. Google Scholar |
[52] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Commun. Math. Phys., 131 (1990), 223.
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.
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.
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.
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.
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.
doi: 10.1007/s005260100105. |
[58] |
P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681.
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.
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.
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, 410 (2000), 192.
|
[62] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, Phys. Rev., 50 (1994), 687.
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.
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.
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.
doi: 10.1007/BF02096642. |
[66] |
Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations,, Nonlinear Differ. Equ. Appl., 19 (2012), 19.
doi: 10.1007/s00030-011-0116-3. |
[67] |
M. Willem, Minimax Theorems,, Birkhäuser, (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.
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.
|
[1] |
Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292 |
[2] |
Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020447 |
[3] |
Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298 |
[4] |
Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125 |
[5] |
Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020436 |
[6] |
Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 |
[7] |
José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020376 |
[8] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
[9] |
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 |
[10] |
Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284 |
[11] |
Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021022 |
[12] |
Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 |
[13] |
Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020461 |
[14] |
Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 |
[15] |
Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020454 |
[16] |
Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020052 |
[17] |
Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020388 |
[18] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020450 |
[19] |
Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 |
[20] |
Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]