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On vector solutions for coupled nonlinear Schrödinger equations with critical exponents

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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).
    Mathematics Subject Classification: Primary: 35A15; Secondary: 35B33, 35B40, 35J50.

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  • [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. ZouPositive 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 $\mathbbR^n$, 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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