# American Institute of Mathematical Sciences

June  2016, 36(6): 3357-3373. doi: 10.3934/dcds.2016.36.3357

## On elliptic systems with Sobolev critical exponent

 1 Department of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China

Received  October 2014 Revised  October 2015 Published  December 2015

We study the following elliptic system with Sobolev critical exponent \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=|u|^{2^*-2}u + \frac{\lambda\alpha}{2^*}|u|^{\alpha-2}|v|^{\beta}u,\, &x\in \mathbb{R}^N, \\ -\Delta v=|v|^{2^*-2}v + \frac{\lambda\beta}{2^*}|u|^{\alpha}|v|^{\beta-2}v,\, &x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $\lambda>0$ is a parameter, $N\geq 3$, $\alpha, \beta>1,$ $\alpha+\beta=2^*:=\frac{2N}{N-2}$, the critical Sobolev exponent. We obtain a uniqueness result on the least energy solutions and show that a manifold of a type of positive solutions is non-degenerate in some ranges of $\lambda,\alpha,\beta,N$ for the above system.
Citation: Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
##### References:
 [1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background,, Phys. Rev. Lett., 82 (1999), 2661. doi: 10.1103/PhysRevLett.82.2661. Google Scholar [2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 453. doi: 10.1016/j.crma.2006.01.024. Google Scholar [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Sci., 75 (2007), 67. doi: 10.1112/jlms/jdl020. Google Scholar [4] T. Bartsch, N. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y. Google Scholar [5] T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator,, Calc. Var. Partial Differential Equations, 18 (2003), 253. doi: 10.1007/s00526-003-0198-9. Google Scholar [6] T. Bartsch, Z. Wang and J. Wei, Bounded states for s coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353. doi: 10.1007/s11784-007-0033-6. Google Scholar [7] G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18. doi: 10.1016/0022-1236(91)90099-Q. Google Scholar [8] W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbbR^N$ with critical growth,, J. Differ. Equ., 252 (2012), 2425. doi: 10.1016/j.jde.2011.09.032. Google Scholar [9] Z. Chen and W. Zou, Positive least energy solutions and phase seperation for coupled Schrödinger equations with critical exponent,, Arch. Ration. Mech. Anal., 205 (2012), 515. doi: 10.1007/s00205-012-0513-8. Google Scholar [10] Z. Chen and W. 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Partial Differ. Equ., 27 (2002), 23. doi: 10.1081/PDE-120002782. Google Scholar [15] B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates,, Phys. Rev. Lett., 78 (1997), 3594. doi: 10.1103/PhysRevLett.78.3594. Google Scholar [16] Y. Huang and D. kang, On the singular elliptic systems involving multiple critical Sobolev exponents,, Nonlinear Anal., 74 (2011), 400. doi: 10.1016/j.na.2010.08.051. Google Scholar [17] S. Kim, On vertor solutions for coupled nonlinear Schrédinger equations with critical exponents,, Comm. Pure Appl. Anal., 12 (2013), 1259. doi: 10.3934/cpaa.2013.12.1259. Google Scholar [18] T. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schréinger equations in $\mathbbR^n$, $n\leq3$,, Comm. Math. Phys., 255 (2005), 629. doi: 10.1007/s00220-005-1313-x. Google Scholar [19] T. Lin and J. Wei, Spikes in two coupled nonlinear Schrédinger equations,, Ann. Inst. H. Poincaré Anal. NonLinéaire, 22 (2005), 403. doi: 10.1016/j.anihpc.2004.03.004. Google Scholar [20] T. Lin and J. Wei, Multiple bound states of nonlinear Schrödinger equations with trapping potentials,, J. Differ. Equ., 229 (2006), 743. Google Scholar [21] Z. Liu and Z. Qang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721. doi: 10.1007/s00220-008-0546-x. Google Scholar [22] C. Menyuk, Nonlinear pulse propagation in birefringent optical fibers,, IEEE. J. Quantum Electron., 23 (1987), 174. doi: 10.1109/JQE.1987.1073308. Google Scholar [23] S. Peng and Z. Wang, Segregated and Synchronized Vector Solutions for Nonlinear Schr\"odinger Systems,, Arch. Ration. Mech. Anal., 208 (2013), 305. doi: 10.1007/s00205-012-0598-0. Google Scholar [24] S. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$,, Soviet Math. Dokl., 6 (1965), 1408. Google Scholar [25] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, Math. Z., 187 (1984), 511. doi: 10.1007/BF01174186. Google Scholar [26] C. Swanson, The best Sobolev constant,, Applicable Anal., 47 (1992), 227. doi: 10.1080/00036819208840142. Google Scholar [27] G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pure Appl., 110 (1976), 353. doi: 10.1007/BF02418013. Google Scholar [28] J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations,, Rend. Lincei mat. Appl., 18 (2007), 279. doi: 10.4171/RLM/495. Google Scholar [29] J. Wei and T. Weth, Asymptotic behavior of solutions of planar systems with strong competition,, Nonlinearity, 21 (2008), 305. doi: 10.1088/0951-7715/21/2/006. Google Scholar [30] J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 37 (2010), 423. doi: 10.1007/s00526-009-0270-1. Google Scholar

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##### References:
 [1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background,, Phys. Rev. Lett., 82 (1999), 2661. doi: 10.1103/PhysRevLett.82.2661. Google Scholar [2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 453. doi: 10.1016/j.crma.2006.01.024. Google Scholar [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Sci., 75 (2007), 67. doi: 10.1112/jlms/jdl020. Google Scholar [4] T. Bartsch, N. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y. Google Scholar [5] T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator,, Calc. Var. Partial Differential Equations, 18 (2003), 253. doi: 10.1007/s00526-003-0198-9. Google Scholar [6] T. Bartsch, Z. Wang and J. Wei, Bounded states for s coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353. doi: 10.1007/s11784-007-0033-6. Google Scholar [7] G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18. doi: 10.1016/0022-1236(91)90099-Q. Google Scholar [8] W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbbR^N$ with critical growth,, J. Differ. Equ., 252 (2012), 2425. doi: 10.1016/j.jde.2011.09.032. Google Scholar [9] Z. Chen and W. Zou, Positive least energy solutions and phase seperation for coupled Schrödinger equations with critical exponent,, Arch. Ration. Mech. Anal., 205 (2012), 515. doi: 10.1007/s00205-012-0513-8. Google Scholar [10] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case,, Calc. Var. Partial Differential Equations, 52 (2015), 423. doi: 10.1007/s00526-014-0717-x. Google Scholar [11] Z. Chen and W. Zou, Existence and symmetry of positive ground states for a doubly critical Schrödinger system,, Trans. Amer. Math. Soc., 367 (2015), 3599. doi: 10.1090/S0002-9947-2014-06237-5. Google Scholar [12] E. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problem,, Math. Ann., 285 (1986), 647. doi: 10.1007/BF01452052. Google Scholar [13] N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutiond for a nonlinear Schrödinger systems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953. doi: 10.1016/j.anihpc.2010.01.009. Google Scholar [14] E. Dancer and S. Yan, Multi-bump solutions for an elliptic problem in expanding domains,, Comm. Partial Differ. Equ., 27 (2002), 23. doi: 10.1081/PDE-120002782. Google Scholar [15] B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates,, Phys. Rev. Lett., 78 (1997), 3594. doi: 10.1103/PhysRevLett.78.3594. Google Scholar [16] Y. Huang and D. kang, On the singular elliptic systems involving multiple critical Sobolev exponents,, Nonlinear Anal., 74 (2011), 400. doi: 10.1016/j.na.2010.08.051. Google Scholar [17] S. Kim, On vertor solutions for coupled nonlinear Schrédinger equations with critical exponents,, Comm. Pure Appl. Anal., 12 (2013), 1259. doi: 10.3934/cpaa.2013.12.1259. Google Scholar [18] T. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schréinger equations in $\mathbbR^n$, $n\leq3$,, Comm. Math. Phys., 255 (2005), 629. doi: 10.1007/s00220-005-1313-x. Google Scholar [19] T. Lin and J. Wei, Spikes in two coupled nonlinear Schrédinger equations,, Ann. Inst. H. Poincaré Anal. NonLinéaire, 22 (2005), 403. doi: 10.1016/j.anihpc.2004.03.004. Google Scholar [20] T. Lin and J. Wei, Multiple bound states of nonlinear Schrödinger equations with trapping potentials,, J. Differ. Equ., 229 (2006), 743. Google Scholar [21] Z. Liu and Z. Qang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721. doi: 10.1007/s00220-008-0546-x. Google Scholar [22] C. Menyuk, Nonlinear pulse propagation in birefringent optical fibers,, IEEE. J. Quantum Electron., 23 (1987), 174. doi: 10.1109/JQE.1987.1073308. Google Scholar [23] S. Peng and Z. Wang, Segregated and Synchronized Vector Solutions for Nonlinear Schr\"odinger Systems,, Arch. Ration. Mech. Anal., 208 (2013), 305. doi: 10.1007/s00205-012-0598-0. Google Scholar [24] S. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$,, Soviet Math. Dokl., 6 (1965), 1408. Google Scholar [25] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, Math. Z., 187 (1984), 511. doi: 10.1007/BF01174186. Google Scholar [26] C. Swanson, The best Sobolev constant,, Applicable Anal., 47 (1992), 227. doi: 10.1080/00036819208840142. Google Scholar [27] G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pure Appl., 110 (1976), 353. doi: 10.1007/BF02418013. Google Scholar [28] J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations,, Rend. Lincei mat. Appl., 18 (2007), 279. doi: 10.4171/RLM/495. Google Scholar [29] J. Wei and T. Weth, Asymptotic behavior of solutions of planar systems with strong competition,, Nonlinearity, 21 (2008), 305. doi: 10.1088/0951-7715/21/2/006. Google Scholar [30] J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 37 (2010), 423. doi: 10.1007/s00526-009-0270-1. Google Scholar
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