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Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents
1. | School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China |
2. | Department of Mathematics, Tsinghua University, Beijing, 100084, China |
$ {\Bbb R}^N: $ |
$ \left\{\begin{array}{ll}-\Delta w+(\lambda a(x)+1)w-(\Delta|w|^2)w = \frac{p}{p+q}|w|^{p-2}w|z|^q+\frac{\alpha}{\alpha+\beta}|w|^{\alpha-2}w|z|^\beta\\\ -\Delta z+(\lambda b(x)+1)z-(\Delta|z|^2)z = \frac{q}{p+q}|w|^p|z|^{q-2}z+\frac{\beta}{\alpha+\beta}|w|^\alpha|z|^{\beta-2}z, \ \end{array}\right. $ |
$ \lambda>0 $ |
$ p>2, q>2, \alpha>2, \beta>2, $ |
$ 2\cdot(2^*-1) < p+q<2\cdot2^* $ |
$ \alpha+ \beta = 2\cdot2^*. $ |
$ \Omega = int \left\{a^{-1}(0)\right\}\cap int \left\{b^{-1}(0)\right\} $ |
$ \lambda $ |
References:
[1] |
S. Adachi, M. Shibata and T. Watanabe,
Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities, Comm. Pure. Appl. Anal., 13 (2014), 97-118.
doi: 10.3934/cpaa.2014.13.97. |
[2] |
C. Alves, D. Filho and M. Sonto,
On systems of elliptic equations involving subcritical or critical Sobolev eponents, Nonlinear Anal., 42 (2000), 771-787.
doi: 10.1016/S0362-546X(99)00121-2. |
[3] |
J. Bezerra do Ó, O. Miyagaki and S. Soares,
Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 248 (2010), 722-744.
doi: 10.1016/j.jde.2009.11.030. |
[4] |
A. Bouard, N. Hayashi and J. Saut,
Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[5] |
H. S. 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.
|
[6] |
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.
|
[7] |
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. |
[8] |
Y. Deng, S. Peng and S. Yan,
Positive solition solutions for generalized quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 37 (2017), 4213-4230.
doi: 10.1016/j.jde.2014.09.006. |
[9] |
Y. Guo and B. Li,
Solutions for qusilinear Schrödinger systems with critical exponents, Z. Angew. Math. Phys., 66 (2015), 517-546.
doi: 10.1007/s00033-014-0416-7. |
[10] |
Y. Guo, X. Liu and F. Zhao,
Positive solutions for quasilinear systems with critical growth, Advan. Nonl. Studies., 13 (2013), 893-919.
doi: 10.1515/ans-2013-0408. |
[11] |
Y. Guo and J. Nie,
Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials, Z. Angew. Math. Phys., 67 (2016), 1-30.
doi: 10.1007/s00033-016-0621-7. |
[12] |
Y. Guo and Z. Tang,
Ground state solutions for the quasilinear Schrödinger systems, J. Math. Anal. Appl., 389 (2012), 322-339.
doi: 10.1016/j.jmaa.2011.11.064. |
[13] |
X. He, A. Qian and W. Zou,
Existence and concentration of positive solutions for quasilinear Schrö dinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.
doi: 10.1088/0951-7715/26/12/3137. |
[14] |
Y. He and G. Li,
Concentrating solition solutions for quasilinear Schrödinger equations involving critical sobolev exponents, Disc. Cont. Dyna. Syst., 36 (2016), 731-762.
doi: 10.3934/dcds.2016.36.731. |
[15] |
L. Jeanjean, T. Lou and Z. Q. Wang,
Multiple normalized solutions for quasi-linear Schrödinger equations, J. Diff. Equat., 259 (2015), 3894-3928.
doi: 10.1016/j.jde.2015.05.008. |
[16] |
J. Liu, X. Liu and Z. Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Part. Diff. Equat., 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[17] |
J. Liu, X. Liu and Z. Q. Wang,
Multibump solutions for quasilinear elliptic equations, Jour. Func. Anal., 262 (2012), 4040-4102.
doi: 10.1016/j.jfa.2012.02.009. |
[18] |
J. Liu, X. Liu and Z. Q. Wang,
Existence theory for quasilinear elliptic equations via regularization approach, Topo. Meth. Nonlinear Anal., 50 (2017), 469-487.
doi: 10.12775/tmna.2017.008. |
[19] |
J. Liu, Y. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrö dinger equation, Ⅱ, J. Diff. Equat., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Liu, Y. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equation via Nehari method, Comm. Part. Diff. Equat., 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[21] |
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. |
[22] |
X. Liu, J. Liu and Z. Q. Wang,
Quasilinear elliptic equation with critical growth via perturbation method, J. Diff. Equat., 254 (2013), 102-124.
doi: 10.1016/j.jde.2012.09.006. |
[23] |
B. Ritchie,
Relativistic self-focusing channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.
|
[24] |
E. Silva and G. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var., 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[25] |
Y. Wang and Y. Shen,
Existence and asymptotic behavior of positive solutions for a class of quasilinear Schrödinger equations, Advan. Nonl. Studies., 1 (2018), 131-150.
doi: 10.1515/ans-2017-6026. |
[26] |
Y. Wang, Y. Zhang and Y. Shen,
Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
[27] |
Y. Wang and W. Zou,
Bound states to critical quasilinear Schrödinger equations, Non. Diff. Equat. App., 19 (2012), 19-47.
doi: 10.1007/s00030-011-0116-3. |
[28] |
M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
K. Wu,
Positive solutons of quasilinear Schrödinger equations critical growth, Appl. Math. Letters., 45 (2015), 52-57.
doi: 10.1016/j.aml.2015.01.005. |
[30] |
X. Zeng, Y. Zhang and H. Zhou,
Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent, Comm. Cont. Math., 16 (2014), 1-32.
doi: 10.1142/S0219199714500345. |
show all references
References:
[1] |
S. Adachi, M. Shibata and T. Watanabe,
Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities, Comm. Pure. Appl. Anal., 13 (2014), 97-118.
doi: 10.3934/cpaa.2014.13.97. |
[2] |
C. Alves, D. Filho and M. Sonto,
On systems of elliptic equations involving subcritical or critical Sobolev eponents, Nonlinear Anal., 42 (2000), 771-787.
doi: 10.1016/S0362-546X(99)00121-2. |
[3] |
J. Bezerra do Ó, O. Miyagaki and S. Soares,
Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 248 (2010), 722-744.
doi: 10.1016/j.jde.2009.11.030. |
[4] |
A. Bouard, N. Hayashi and J. Saut,
Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[5] |
H. S. 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.
|
[6] |
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.
|
[7] |
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. |
[8] |
Y. Deng, S. Peng and S. Yan,
Positive solition solutions for generalized quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 37 (2017), 4213-4230.
doi: 10.1016/j.jde.2014.09.006. |
[9] |
Y. Guo and B. Li,
Solutions for qusilinear Schrödinger systems with critical exponents, Z. Angew. Math. Phys., 66 (2015), 517-546.
doi: 10.1007/s00033-014-0416-7. |
[10] |
Y. Guo, X. Liu and F. Zhao,
Positive solutions for quasilinear systems with critical growth, Advan. Nonl. Studies., 13 (2013), 893-919.
doi: 10.1515/ans-2013-0408. |
[11] |
Y. Guo and J. Nie,
Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials, Z. Angew. Math. Phys., 67 (2016), 1-30.
doi: 10.1007/s00033-016-0621-7. |
[12] |
Y. Guo and Z. Tang,
Ground state solutions for the quasilinear Schrödinger systems, J. Math. Anal. Appl., 389 (2012), 322-339.
doi: 10.1016/j.jmaa.2011.11.064. |
[13] |
X. He, A. Qian and W. Zou,
Existence and concentration of positive solutions for quasilinear Schrö dinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.
doi: 10.1088/0951-7715/26/12/3137. |
[14] |
Y. He and G. Li,
Concentrating solition solutions for quasilinear Schrödinger equations involving critical sobolev exponents, Disc. Cont. Dyna. Syst., 36 (2016), 731-762.
doi: 10.3934/dcds.2016.36.731. |
[15] |
L. Jeanjean, T. Lou and Z. Q. Wang,
Multiple normalized solutions for quasi-linear Schrödinger equations, J. Diff. Equat., 259 (2015), 3894-3928.
doi: 10.1016/j.jde.2015.05.008. |
[16] |
J. Liu, X. Liu and Z. Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Part. Diff. Equat., 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[17] |
J. Liu, X. Liu and Z. Q. Wang,
Multibump solutions for quasilinear elliptic equations, Jour. Func. Anal., 262 (2012), 4040-4102.
doi: 10.1016/j.jfa.2012.02.009. |
[18] |
J. Liu, X. Liu and Z. Q. Wang,
Existence theory for quasilinear elliptic equations via regularization approach, Topo. Meth. Nonlinear Anal., 50 (2017), 469-487.
doi: 10.12775/tmna.2017.008. |
[19] |
J. Liu, Y. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrö dinger equation, Ⅱ, J. Diff. Equat., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Liu, Y. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equation via Nehari method, Comm. Part. Diff. Equat., 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[21] |
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. |
[22] |
X. Liu, J. Liu and Z. Q. Wang,
Quasilinear elliptic equation with critical growth via perturbation method, J. Diff. Equat., 254 (2013), 102-124.
doi: 10.1016/j.jde.2012.09.006. |
[23] |
B. Ritchie,
Relativistic self-focusing channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.
|
[24] |
E. Silva and G. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var., 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[25] |
Y. Wang and Y. Shen,
Existence and asymptotic behavior of positive solutions for a class of quasilinear Schrödinger equations, Advan. Nonl. Studies., 1 (2018), 131-150.
doi: 10.1515/ans-2017-6026. |
[26] |
Y. Wang, Y. Zhang and Y. Shen,
Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
[27] |
Y. Wang and W. Zou,
Bound states to critical quasilinear Schrödinger equations, Non. Diff. Equat. App., 19 (2012), 19-47.
doi: 10.1007/s00030-011-0116-3. |
[28] |
M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
K. Wu,
Positive solutons of quasilinear Schrödinger equations critical growth, Appl. Math. Letters., 45 (2015), 52-57.
doi: 10.1016/j.aml.2015.01.005. |
[30] |
X. Zeng, Y. Zhang and H. Zhou,
Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent, Comm. Cont. Math., 16 (2014), 1-32.
doi: 10.1142/S0219199714500345. |
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