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Equivalence of sharp Trudinger-Moser-Adams Inequalities
Positive ground state solutions of a quadratically coupled schrödinger system
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China |
$\begin{equation*}\left\{\begin{array}{ll}-\Delta u+\lambda_1u=\mu_1u^2+2\alpha uv+\gamma v^2, & \mbox{in }\Omega,\\-\Delta v+\lambda_2v=\mu_2v^2+2\gamma uv+\alpha u^2, & \mbox{in }\Omega,\\u=v=0, & \mbox{on }\partial\Omega,\end{array}\right.\end{equation*}$ |
References:
[1] |
A. Ambrosetti and E. Colorado,
Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[2] |
T. Bartsch, N. Dancer and Z.-Q. Wang,
A Liouville theorem, a-priori bounds and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[3] |
T. Bartsch, Z.-Q. Wang and J. Wei,
Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[4] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[5] |
A. V. Buryak, P. Di Trapani, D. V. Skryabin and S. Trillo,
Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235.
doi: 10.1016/S0370-1573(02)00196-5. |
[6] |
A. V. Buryak and Y. S. Kivshar,
Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412.
doi: 10.1016/0375-9601(94)00989-3. |
[7] |
Z. Chen and W. Zou,
Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107.
doi: 10.1016/j.jfa.2012.01.001. |
[8] |
Z. Chen and W. Zou,
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.
doi: 10.1007/s00205-012-0513-8. |
[9] |
Z. Chen and W. Zou,
An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations, 48 (2013), 695-711.
doi: 10.1007/s00526-012-0568-2. |
[10] |
Z. Chen and W. Zou,
Standing waves for linearly coupled Schrödinger equations with critical exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 429-447.
doi: 10.1016/j.anihpc.2013.04.003. |
[11] |
E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.
doi: 10.1007/978-3-642-61798-0. |
[13] |
Z. Guo,
Positive ground state solutions for a nonlinearly coupled Schrödinger system with critical exponents in ℝ4, J. Math. Anal. Appl., 430 (2015), 950-970.
doi: 10.1016/j.jmaa.2015.05.037. |
[14] |
C. T.-Lin and J. Wei,
Ground state of N coupled nonlinear Schrödinger equations in ℝn. n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[15] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[16] |
M. Willem,
Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications. 24, Birkhäuser Boston, Inc., Boston, MA, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[17] |
A. C. Yew, A. R. Champneys and P. J. McKenna,
Multiple solitary waves due to secondharmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52.
doi: 10.1007/s003329900063. |
[18] |
A. C. Yew,
Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137.
doi: 10.1006/jdeq.2000.3922. |
[19] |
L. Zhao, F. Zhao and J. Shi,
Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691.
doi: 10.1007/s00526-015-0879-1. |
show all references
References:
[1] |
A. Ambrosetti and E. Colorado,
Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[2] |
T. Bartsch, N. Dancer and Z.-Q. Wang,
A Liouville theorem, a-priori bounds and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[3] |
T. Bartsch, Z.-Q. Wang and J. Wei,
Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[4] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[5] |
A. V. Buryak, P. Di Trapani, D. V. Skryabin and S. Trillo,
Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235.
doi: 10.1016/S0370-1573(02)00196-5. |
[6] |
A. V. Buryak and Y. S. Kivshar,
Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412.
doi: 10.1016/0375-9601(94)00989-3. |
[7] |
Z. Chen and W. Zou,
Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107.
doi: 10.1016/j.jfa.2012.01.001. |
[8] |
Z. Chen and W. Zou,
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.
doi: 10.1007/s00205-012-0513-8. |
[9] |
Z. Chen and W. Zou,
An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations, 48 (2013), 695-711.
doi: 10.1007/s00526-012-0568-2. |
[10] |
Z. Chen and W. Zou,
Standing waves for linearly coupled Schrödinger equations with critical exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 429-447.
doi: 10.1016/j.anihpc.2013.04.003. |
[11] |
E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.
doi: 10.1007/978-3-642-61798-0. |
[13] |
Z. Guo,
Positive ground state solutions for a nonlinearly coupled Schrödinger system with critical exponents in ℝ4, J. Math. Anal. Appl., 430 (2015), 950-970.
doi: 10.1016/j.jmaa.2015.05.037. |
[14] |
C. T.-Lin and J. Wei,
Ground state of N coupled nonlinear Schrödinger equations in ℝn. n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[15] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[16] |
M. Willem,
Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications. 24, Birkhäuser Boston, Inc., Boston, MA, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[17] |
A. C. Yew, A. R. Champneys and P. J. McKenna,
Multiple solitary waves due to secondharmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52.
doi: 10.1007/s003329900063. |
[18] |
A. C. Yew,
Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137.
doi: 10.1006/jdeq.2000.3922. |
[19] |
L. Zhao, F. Zhao and J. Shi,
Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691.
doi: 10.1007/s00526-015-0879-1. |
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