# American Institute of Mathematical Sciences

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 & 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 Poincar\'e, 27 (2010), 953. 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. 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. 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. 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. 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. doi: 10.1109/JQE.1987.1073308. Google Scholar [20] G. Talenti, Best constants in Sobolev inequality,, Annali di Mat., 110 (1976), 353. 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. 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. 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). 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. doi: 10.1007/s00205-008-0121-9. Google Scholar [25] M. Willem, "Minimax Theorems,", PNLDE 24, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 Poincar\'e, 27 (2010), 953. 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. 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. 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. 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. 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. doi: 10.1109/JQE.1987.1073308. Google Scholar [20] G. Talenti, Best constants in Sobolev inequality,, Annali di Mat., 110 (1976), 353. 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. 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. 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). 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. doi: 10.1007/s00205-008-0121-9. Google Scholar [25] M. Willem, "Minimax Theorems,", PNLDE 24, (1996). Google Scholar
 [1] A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419 [2] Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991 [3] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [4] Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205 [5] Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 [6] Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789 [7] M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982 [8] Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911 [9] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [10] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [11] Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 [12] Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 [13] Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118 [14] Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 [15] Myeongju Chae, Sunggeum Hong, Sanghyuk Lee. Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 909-928. doi: 10.3934/dcds.2011.29.909 [16] Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921 [17] Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883 [18] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389 [19] Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377 [20] Alessio Pomponio, Simone Secchi. A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities. Communications on Pure & Applied Analysis, 2010, 9 (3) : 741-750. doi: 10.3934/cpaa.2010.9.741

2018 Impact Factor: 0.925