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May  2013, 12(3): 1259-1277. doi: 10.3934/cpaa.2013.12.1259

On vector solutions for coupled nonlinear Schrödinger equations with critical exponents

Seunghyeok Kim 1,

1. 

Department of Mathematics, Pohang University of Science and Technology, Pohang, Kyungbuk, South Korea

Received  December 2011 Revised  June 2012 Published  September 2012

In this paper, we study the existence and asymptotic behavior of a solution with positive components (which we call a vector solution) for the coupled system of nonlinear Schrödinger equations with doubly critical exponents \begin{eqnarray*} \Delta u + \lambda_1 u + \mu_1 u^{\frac{N+2}{N-2}} + \beta u^{\frac{2}{N-2}}v^{\frac{N}{N-2}} = 0\\ \Delta v + \lambda_2 v + \mu_2 v^{\frac{N+2}{N-2}} + \beta u^{\frac{N}{N-2}}v^{\frac{2}{N-2}} = 0 \quad in \quad \Omega\\ u, v > 0 \quad in \quad \Omega, \quad u, v = 0 \quad on \quad \partial \Omega \end{eqnarray*} as the coupling coefficient $\beta \in R$ tends to 0 or $+\infty$, where the domain $\Omega \subset R^n (N \geq 3)$ is smooth bounded and certain conditions on $\lambda_1, \lambda_2 > 0$ and $\mu_1, \mu_2 > 0$ are imposed. This system naturally arises as a counterpart of the Brezis-Nirenberg problem (Comm. Pure Appl. Math. 36: 437-477, 1983).
Keywords: Coupled nonlinear Schrödinger equations, critical exponent, Nehari manifold.
Mathematics Subject Classification: Primary: 35A15; Secondary: 35B33, 35B40, 35J5.
Citation: Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Diff. Eqs., 19 (2006), 200-207.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent,, preprint., ().   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[20]

G. Talenti, Best constants in Sobolev inequality, Annali di Mat., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[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.  Google Scholar

[22]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Commum. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. Willem, "Minimax Theorems," PNLDE 24, Birkhäuser, 1996  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Diff. Eqs., 19 (2006), 200-207.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

G. Talenti, Best constants in Sobolev inequality, Annali di Mat., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[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.  Google Scholar

[22]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Commum. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. Willem, "Minimax Theorems," PNLDE 24, Birkhäuser, 1996  Google Scholar

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