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Infinite multiplicity for an inhomogeneous supercritical problem in entire space
On vector solutions for coupled nonlinear Schrödinger equations with critical exponents
1. | Department of Mathematics, Pohang University of Science and Technology, Pohang, Kyungbuk, South Korea |
References:
[1] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[2] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $R^n$, Calc. Var., 34 (2009), 97-137.
doi: 10.1007/s00526-008-0177-2. |
[4] |
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., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Diff. Eqs., 19 (2006), 200-207. |
[6] |
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. |
[7] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[8] |
H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. |
[9] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[10] |
H. Brezis 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. |
[11] |
J. Byeon, Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial Diff. Eq., 22 (1997), 1731-1769.
doi: 10.1080/03605309708821317. |
[12] |
J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal., 185 (2007), 185-200.
doi: 10.1007/s00205-006-0019-3. |
[13] |
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, preprint. |
[14] |
E. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[15] |
B. D. Esry, C. H. Greene, Jr. J. P. Burke and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[16] |
T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n \leq 3$, Commum. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[17] |
Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear schrodinger systems, Commum. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[18] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Eq., 229 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[19] |
C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron., 23 (1987), 174-176.
doi: 10.1109/JQE.1987.1073308. |
[20] |
G. Talenti, Best constants in Sobolev inequality, Annali di Mat., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[21] |
S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates, Arch. Rational Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[22] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Commum. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[23] |
G. M. Wei and Y. H. Wang, Existence of least energy solutions to coupled elliptic systems with critical nonlinearities, Electron. J. Diff. Eq., 49 (2008), 8 pp. |
[24] |
J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
[25] |
show all references
References:
[1] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[2] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $R^n$, Calc. Var., 34 (2009), 97-137.
doi: 10.1007/s00526-008-0177-2. |
[4] |
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., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Diff. Eqs., 19 (2006), 200-207. |
[6] |
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. |
[7] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[8] |
H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. |
[9] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[10] |
H. Brezis 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. |
[11] |
J. Byeon, Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial Diff. Eq., 22 (1997), 1731-1769.
doi: 10.1080/03605309708821317. |
[12] |
J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal., 185 (2007), 185-200.
doi: 10.1007/s00205-006-0019-3. |
[13] |
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, preprint. |
[14] |
E. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[15] |
B. D. Esry, C. H. Greene, Jr. J. P. Burke and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[16] |
T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n \leq 3$, Commum. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[17] |
Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear schrodinger systems, Commum. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[18] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Eq., 229 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[19] |
C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron., 23 (1987), 174-176.
doi: 10.1109/JQE.1987.1073308. |
[20] |
G. Talenti, Best constants in Sobolev inequality, Annali di Mat., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[21] |
S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates, Arch. Rational Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[22] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Commum. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[23] |
G. M. Wei and Y. H. Wang, Existence of least energy solutions to coupled elliptic systems with critical nonlinearities, Electron. J. Diff. Eq., 49 (2008), 8 pp. |
[24] |
J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
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