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

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

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).
Citation: Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and 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.

[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]

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

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.

[25]

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

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