On vector solutions for coupled nonlinear Schrödinger equations with critical exponents

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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

Abstract
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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 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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