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On two-sided estimates for the nonlinear Fourier transform of KdV
On elliptic systems with Sobolev critical exponent
1. | Department of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China |
References:
[1] |
N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.
doi: 10.1103/PhysRevLett.82.2661. |
[2] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[3] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Sci., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[4] |
T. Bartsch, N. Dancer and Z. 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. |
[5] |
T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differential Equations, 18 (2003), 253-268.
doi: 10.1007/s00526-003-0198-9. |
[6] |
T. Bartsch, Z. Wang and J. Wei, Bounded states for s coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[7] |
G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.
doi: 10.1016/0022-1236(91)90099-Q. |
[8] |
W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N2$ with critical growth, J. Differ. Equ., 252 (2012), 2425-2447.
doi: 10.1016/j.jde.2011.09.032. |
[9] |
Z. Chen and W. Zou, Positive least energy solutions and phase seperation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.
doi: 10.1007/s00205-012-0513-8. |
[10] |
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.
doi: 10.1007/s00526-014-0717-x. |
[11] |
Z. Chen and W. Zou, Existence and symmetry of positive ground states for a doubly critical Schrödinger system, Trans. Amer. Math. Soc., 367 (2015), 3599-3646.
doi: 10.1090/S0002-9947-2014-06237-5. |
[12] |
E. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problem, Math. Ann., 285 (1986), 647-669.
doi: 10.1007/BF01452052. |
[13] |
N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutiond for a nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[14] |
E. Dancer and S. Yan, Multi-bump solutions for an elliptic problem in expanding domains, Comm. Partial Differ. Equ., 27 (2002), 23-55.
doi: 10.1081/PDE-120002782. |
[15] |
B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[16] |
Y. Huang and D. kang, On the singular elliptic systems involving multiple critical Sobolev exponents, Nonlinear Anal., 74 (2011), 400-412.
doi: 10.1016/j.na.2010.08.051. |
[17] |
S. Kim, On vertor solutions for coupled nonlinear Schrédinger equations with critical exponents, Comm. Pure Appl. Anal., 12 (2013), 1259-1277.
doi: 10.3934/cpaa.2013.12.1259. |
[18] |
T. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schréinger equations in $\mathbb{R}^{N}$, $n\leq3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[19] |
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrédinger equations, Ann. Inst. H. Poincaré Anal. NonLinéaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[20] |
T. Lin and J. Wei, Multiple bound states of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 743-767. |
[21] |
Z. Liu and Z. Qang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[22] |
C. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE. J. Quantum Electron., 23 (1987), 174-176.
doi: 10.1109/JQE.1987.1073308. |
[23] |
S. Peng and Z. Wang, Segregated and Synchronized Vector Solutions for Nonlinear Schr\"odinger Systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[24] |
S. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Soviet Math. Dokl., 6 (1965), 1408-1411. |
[25] |
M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[26] |
C. Swanson, The best Sobolev constant, Applicable Anal., 47 (1992), 227-239.
doi: 10.1080/00036819208840142. |
[27] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pure Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[28] |
J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
[29] |
J. Wei and T. Weth, Asymptotic behavior of solutions of planar systems with strong competition, Nonlinearity, 21 (2008), 305-317.
doi: 10.1088/0951-7715/21/2/006. |
[30] |
J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N2$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.
doi: 10.1007/s00526-009-0270-1. |
show all references
References:
[1] |
N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.
doi: 10.1103/PhysRevLett.82.2661. |
[2] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[3] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Sci., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[4] |
T. Bartsch, N. Dancer and Z. 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. |
[5] |
T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differential Equations, 18 (2003), 253-268.
doi: 10.1007/s00526-003-0198-9. |
[6] |
T. Bartsch, Z. Wang and J. Wei, Bounded states for s coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[7] |
G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.
doi: 10.1016/0022-1236(91)90099-Q. |
[8] |
W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N2$ with critical growth, J. Differ. Equ., 252 (2012), 2425-2447.
doi: 10.1016/j.jde.2011.09.032. |
[9] |
Z. Chen and W. Zou, Positive least energy solutions and phase seperation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.
doi: 10.1007/s00205-012-0513-8. |
[10] |
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.
doi: 10.1007/s00526-014-0717-x. |
[11] |
Z. Chen and W. Zou, Existence and symmetry of positive ground states for a doubly critical Schrödinger system, Trans. Amer. Math. Soc., 367 (2015), 3599-3646.
doi: 10.1090/S0002-9947-2014-06237-5. |
[12] |
E. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problem, Math. Ann., 285 (1986), 647-669.
doi: 10.1007/BF01452052. |
[13] |
N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutiond for a nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[14] |
E. Dancer and S. Yan, Multi-bump solutions for an elliptic problem in expanding domains, Comm. Partial Differ. Equ., 27 (2002), 23-55.
doi: 10.1081/PDE-120002782. |
[15] |
B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[16] |
Y. Huang and D. kang, On the singular elliptic systems involving multiple critical Sobolev exponents, Nonlinear Anal., 74 (2011), 400-412.
doi: 10.1016/j.na.2010.08.051. |
[17] |
S. Kim, On vertor solutions for coupled nonlinear Schrédinger equations with critical exponents, Comm. Pure Appl. Anal., 12 (2013), 1259-1277.
doi: 10.3934/cpaa.2013.12.1259. |
[18] |
T. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schréinger equations in $\mathbb{R}^{N}$, $n\leq3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[19] |
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrédinger equations, Ann. Inst. H. Poincaré Anal. NonLinéaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[20] |
T. Lin and J. Wei, Multiple bound states of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 743-767. |
[21] |
Z. Liu and Z. Qang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[22] |
C. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE. J. Quantum Electron., 23 (1987), 174-176.
doi: 10.1109/JQE.1987.1073308. |
[23] |
S. Peng and Z. Wang, Segregated and Synchronized Vector Solutions for Nonlinear Schr\"odinger Systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[24] |
S. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Soviet Math. Dokl., 6 (1965), 1408-1411. |
[25] |
M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[26] |
C. Swanson, The best Sobolev constant, Applicable Anal., 47 (1992), 227-239.
doi: 10.1080/00036819208840142. |
[27] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pure Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[28] |
J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
[29] |
J. Wei and T. Weth, Asymptotic behavior of solutions of planar systems with strong competition, Nonlinearity, 21 (2008), 305-317.
doi: 10.1088/0951-7715/21/2/006. |
[30] |
J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N2$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.
doi: 10.1007/s00526-009-0270-1. |
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