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Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $
1. | Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China |
2. | Department of Mathematics, Shaoyang University, Shaoyang, Hunan, 422000, China |
$ -\Delta u+V(x)u-\Delta(u^2)u = a(\epsilon x)g(u), \; \; \; \; x\in\mathbb R^N, $ |
$ V(x)>0 $ |
$ u>0 $ |
$ a $ |
$ g $ |
$ a $ |
$ m $ |
$ m $ |
References:
[1] |
C. O. Alves, G. M. Figueiredo and J. A. Santos,
Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435-456.
doi: 10.12775/TMNA.2014.055. |
[2] |
C. O. Alves and G. M. Figueiredo,
On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $\mathbb R^N$, J. Differential Equations, 246 (2009), 1288-1311.
doi: 10.1016/j.jde.2008.08.004. |
[3] |
C. O. Alves and C. Torres Ledesma, Existence and multiplicity of solutions for a non-linear Schrödinger equation with non-local regional diffusion, J. Math. Phys., 58 (2017), 111507, 19pp.
doi: 10.1063/1.5011724. |
[4] |
A. Borovskiik and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1983), 562-573. Google Scholar |
[5] |
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 B, 5 (1993), 3539-3550. Google Scholar |
[6] |
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. |
[7] |
K. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[8] |
D. Cassani, J. M. Bezerra do Ó and A. Moameni,
Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 9 (2010), 281-306.
doi: 10.3934/cpaa.2010.9.281. |
[9] |
D. M. Cao and E. S. Noussair,
Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb R^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 567-588.
doi: 10.1016/S0294-1449(16)30115-9. |
[10] |
D. M. Cao and H. S. Zhou,
Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463.
doi: 10.1017/S0308210500022836. |
[11] |
P. C. Carrião, R. Lehrer and O. H. Miyagaki,
Existence of solutions to a class of asymptotically linear Schrödinger equations in $\mathbb R^N$ via the Pohozaev manifold, J. Math. Anal. Appl., 428 (2015), 165-183.
doi: 10.1016/j.jmaa.2015.02.060. |
[12] |
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. |
[13] |
F. Esposito, A. Farina and B. Sciunzi,
Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983.
doi: 10.1016/j.jde.2018.04.030. |
[14] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.
doi: 10.1016/j.jde.2012.11.017. |
[15] |
E. Gloss,
Existence and concentration of positive solution for a quasilinear equation in $\mathbb R^N$, J. Math. Anal. Appl., 371 (2010), 465-484.
doi: 10.1016/j.jmaa.2010.05.033. |
[16] |
X. He, A. Qian and W. Zou,
Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.
doi: 10.1088/0951-7715/26/12/3137. |
[17] |
T. S. Hsu, H. L. Lin and C. C. Hu,
Multiple positive solutions of quasilinear elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 388 (2012), 500-512.
doi: 10.1016/j.jmaa.2011.11.010. |
[18] |
S. Kurihura,
Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3801-3805.
doi: 10.1143/JPSJ.50.3801. |
[19] |
J. Q. Liu, Y. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equation, II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equations, I. Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[21] |
J. Liu, S. Chen and X. Wu,
Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 395 (2012), 608-615.
doi: 10.1016/j.jmaa.2012.05.063. |
[22] |
X. Q. Liu, J. Q. 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. |
[23] |
X. Q. Liu, J. Q. 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. |
[24] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[25] |
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. |
[26] |
A. Moameni,
Existence of soliton solutions for a quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$, Commun. Pure Appl. Anal., 7 (2008), 89-105.
doi: 10.3934/cpaa.2008.7.89. |
[27] |
K. Mahiout and C. O. Alves,
Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces, Complex Var. Elliptic Equ., 62 (2016), 767-785.
doi: 10.1080/17476933.2016.1243669. |
[28] |
O. H. Miyagaki and S. I. Moreira,
Nonnegative solution for quasilinear Schrödinger equations that include supercritical exponents with nonlinearities that are indefinite in sign, J. Math. Anal. Appl., 421 (2015), 643-655.
doi: 10.1016/j.jmaa.2014.06.074. |
[29] |
Y. Miyamoto and Y. Naito,
Singular extremal solutions for supercritical elliptic equations in a ball, J. Differential Equations, 265 (2018), 2842-2885.
doi: 10.1016/j.jde.2018.04.055. |
[30] |
Q. H. Miyagaki and S. I. Moreira,
Nonnegative solution for quasilinear Schrödinger equations involving supercritical exponent with nonlinearities indefinitie in sign, J. Math. Anal. Appl., 421 (2015), 643-655.
doi: 10.1016/j.jmaa.2014.06.074. |
[31] |
M. Poppenbery, 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. |
[32] |
M. D. Pina 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. |
[33] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[34] |
E. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.
doi: 10.1016/j.na.2009.11.037. |
[35] |
W. Wang, X. Yang and F. K. Zhao,
Existence and concentration of ground states to a quasilinear problem with competing potentials, Nonlinear Anal., 102 (2014), 120-132.
doi: 10.1016/j.na.2014.01.025. |
[36] |
Y. J. Wang, Y. M. Zhang and Y. T. Shen,
Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comput., 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
[37] |
M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[38] |
X. Wu and K. Wu, Existence of positive solutions, negative solutions and high energy solutions for quasilinear elliptic equations on $\mathbb R^N$, Nonlinear Anal. Real World Appl., 16 (2014), 48-64. Google Scholar |
[39] |
J. Zhang, X. Tang and W. Zhang,
Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl., 420 (2014), 1762-1775.
doi: 10.1016/j.jmaa.2014.06.055. |
show all references
References:
[1] |
C. O. Alves, G. M. Figueiredo and J. A. Santos,
Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435-456.
doi: 10.12775/TMNA.2014.055. |
[2] |
C. O. Alves and G. M. Figueiredo,
On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $\mathbb R^N$, J. Differential Equations, 246 (2009), 1288-1311.
doi: 10.1016/j.jde.2008.08.004. |
[3] |
C. O. Alves and C. Torres Ledesma, Existence and multiplicity of solutions for a non-linear Schrödinger equation with non-local regional diffusion, J. Math. Phys., 58 (2017), 111507, 19pp.
doi: 10.1063/1.5011724. |
[4] |
A. Borovskiik and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1983), 562-573. Google Scholar |
[5] |
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 B, 5 (1993), 3539-3550. Google Scholar |
[6] |
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. |
[7] |
K. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[8] |
D. Cassani, J. M. Bezerra do Ó and A. Moameni,
Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 9 (2010), 281-306.
doi: 10.3934/cpaa.2010.9.281. |
[9] |
D. M. Cao and E. S. Noussair,
Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb R^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 567-588.
doi: 10.1016/S0294-1449(16)30115-9. |
[10] |
D. M. Cao and H. S. Zhou,
Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463.
doi: 10.1017/S0308210500022836. |
[11] |
P. C. Carrião, R. Lehrer and O. H. Miyagaki,
Existence of solutions to a class of asymptotically linear Schrödinger equations in $\mathbb R^N$ via the Pohozaev manifold, J. Math. Anal. Appl., 428 (2015), 165-183.
doi: 10.1016/j.jmaa.2015.02.060. |
[12] |
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. |
[13] |
F. Esposito, A. Farina and B. Sciunzi,
Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983.
doi: 10.1016/j.jde.2018.04.030. |
[14] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.
doi: 10.1016/j.jde.2012.11.017. |
[15] |
E. Gloss,
Existence and concentration of positive solution for a quasilinear equation in $\mathbb R^N$, J. Math. Anal. Appl., 371 (2010), 465-484.
doi: 10.1016/j.jmaa.2010.05.033. |
[16] |
X. He, A. Qian and W. Zou,
Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.
doi: 10.1088/0951-7715/26/12/3137. |
[17] |
T. S. Hsu, H. L. Lin and C. C. Hu,
Multiple positive solutions of quasilinear elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 388 (2012), 500-512.
doi: 10.1016/j.jmaa.2011.11.010. |
[18] |
S. Kurihura,
Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3801-3805.
doi: 10.1143/JPSJ.50.3801. |
[19] |
J. Q. Liu, Y. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equation, II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equations, I. Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[21] |
J. Liu, S. Chen and X. Wu,
Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 395 (2012), 608-615.
doi: 10.1016/j.jmaa.2012.05.063. |
[22] |
X. Q. Liu, J. Q. 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. |
[23] |
X. Q. Liu, J. Q. 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. |
[24] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[25] |
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. |
[26] |
A. Moameni,
Existence of soliton solutions for a quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$, Commun. Pure Appl. Anal., 7 (2008), 89-105.
doi: 10.3934/cpaa.2008.7.89. |
[27] |
K. Mahiout and C. O. Alves,
Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces, Complex Var. Elliptic Equ., 62 (2016), 767-785.
doi: 10.1080/17476933.2016.1243669. |
[28] |
O. H. Miyagaki and S. I. Moreira,
Nonnegative solution for quasilinear Schrödinger equations that include supercritical exponents with nonlinearities that are indefinite in sign, J. Math. Anal. Appl., 421 (2015), 643-655.
doi: 10.1016/j.jmaa.2014.06.074. |
[29] |
Y. Miyamoto and Y. Naito,
Singular extremal solutions for supercritical elliptic equations in a ball, J. Differential Equations, 265 (2018), 2842-2885.
doi: 10.1016/j.jde.2018.04.055. |
[30] |
Q. H. Miyagaki and S. I. Moreira,
Nonnegative solution for quasilinear Schrödinger equations involving supercritical exponent with nonlinearities indefinitie in sign, J. Math. Anal. Appl., 421 (2015), 643-655.
doi: 10.1016/j.jmaa.2014.06.074. |
[31] |
M. Poppenbery, 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. |
[32] |
M. D. Pina 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. |
[33] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[34] |
E. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.
doi: 10.1016/j.na.2009.11.037. |
[35] |
W. Wang, X. Yang and F. K. Zhao,
Existence and concentration of ground states to a quasilinear problem with competing potentials, Nonlinear Anal., 102 (2014), 120-132.
doi: 10.1016/j.na.2014.01.025. |
[36] |
Y. J. Wang, Y. M. Zhang and Y. T. Shen,
Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comput., 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
[37] |
M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[38] |
X. Wu and K. Wu, Existence of positive solutions, negative solutions and high energy solutions for quasilinear elliptic equations on $\mathbb R^N$, Nonlinear Anal. Real World Appl., 16 (2014), 48-64. Google Scholar |
[39] |
J. Zhang, X. Tang and W. Zhang,
Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl., 420 (2014), 1762-1775.
doi: 10.1016/j.jmaa.2014.06.055. |
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