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 and Continuous Dynamical Systems, 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), 2661-2664. doi: 10.1103/PhysRevLett.82.2661.

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024.

[3]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Sci., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.

[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), 345-361. doi: 10.1007/s00526-009-0265-y.

[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), 253-268. doi: 10.1007/s00526-003-0198-9.

[6]

T. Bartsch, Z. Wang and J. Wei, Bounded states for s coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.

[7]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q.

[8]

W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N2$ with critical growth, J. Differ. Equ., 252 (2012), 2425-2447. doi: 10.1016/j.jde.2011.09.032.

[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), 515-551. doi: 10.1007/s00205-012-0513-8.

[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), 423-467. doi: 10.1007/s00526-014-0717-x.

[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), 3599-3646. doi: 10.1090/S0002-9947-2014-06237-5.

[12]

E. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problem, Math. Ann., 285 (1986), 647-669. doi: 10.1007/BF01452052.

[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), 953-969. doi: 10.1016/j.anihpc.2010.01.009.

[14]

E. Dancer and S. Yan, Multi-bump solutions for an elliptic problem in expanding domains, Comm. Partial Differ. Equ., 27 (2002), 23-55. doi: 10.1081/PDE-120002782.

[15]

B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. doi: 10.1103/PhysRevLett.78.3594.

[16]

Y. Huang and D. kang, On the singular elliptic systems involving multiple critical Sobolev exponents, Nonlinear Anal., 74 (2011), 400-412. doi: 10.1016/j.na.2010.08.051.

[17]

S. Kim, On vertor solutions for coupled nonlinear Schrédinger equations with critical exponents, Comm. Pure Appl. Anal., 12 (2013), 1259-1277. doi: 10.3934/cpaa.2013.12.1259.

[18]

T. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schréinger equations in $\mathbb{R}^{N}$, $n\leq3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[19]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrédinger equations, Ann. Inst. H. Poincaré Anal. NonLinéaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004.

[20]

T. Lin and J. Wei, Multiple bound states of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 743-767.

[21]

Z. Liu and Z. Qang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.

[22]

C. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE. J. Quantum Electron., 23 (1987), 174-176. doi: 10.1109/JQE.1987.1073308.

[23]

S. Peng and Z. Wang, Segregated and Synchronized Vector Solutions for Nonlinear Schr\"odinger Systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0.

[24]

S. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Soviet Math. Dokl., 6 (1965), 1408-1411.

[25]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.

[26]

C. Swanson, The best Sobolev constant, Applicable Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.

[27]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pure Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[28]

J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei mat. Appl., 18 (2007), 279-293. doi: 10.4171/RLM/495.

[29]

J. Wei and T. Weth, Asymptotic behavior of solutions of planar systems with strong competition, Nonlinearity, 21 (2008), 305-317. doi: 10.1088/0951-7715/21/2/006.

[30]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N2$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1.

show all references

References:
[1]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664. doi: 10.1103/PhysRevLett.82.2661.

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024.

[3]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Sci., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.

[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), 345-361. doi: 10.1007/s00526-009-0265-y.

[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), 253-268. doi: 10.1007/s00526-003-0198-9.

[6]

T. Bartsch, Z. Wang and J. Wei, Bounded states for s coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.

[7]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q.

[8]

W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N2$ with critical growth, J. Differ. Equ., 252 (2012), 2425-2447. doi: 10.1016/j.jde.2011.09.032.

[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), 515-551. doi: 10.1007/s00205-012-0513-8.

[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), 423-467. doi: 10.1007/s00526-014-0717-x.

[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), 3599-3646. doi: 10.1090/S0002-9947-2014-06237-5.

[12]

E. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problem, Math. Ann., 285 (1986), 647-669. doi: 10.1007/BF01452052.

[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), 953-969. doi: 10.1016/j.anihpc.2010.01.009.

[14]

E. Dancer and S. Yan, Multi-bump solutions for an elliptic problem in expanding domains, Comm. Partial Differ. Equ., 27 (2002), 23-55. doi: 10.1081/PDE-120002782.

[15]

B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. doi: 10.1103/PhysRevLett.78.3594.

[16]

Y. Huang and D. kang, On the singular elliptic systems involving multiple critical Sobolev exponents, Nonlinear Anal., 74 (2011), 400-412. doi: 10.1016/j.na.2010.08.051.

[17]

S. Kim, On vertor solutions for coupled nonlinear Schrédinger equations with critical exponents, Comm. Pure Appl. Anal., 12 (2013), 1259-1277. doi: 10.3934/cpaa.2013.12.1259.

[18]

T. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schréinger equations in $\mathbb{R}^{N}$, $n\leq3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[19]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrédinger equations, Ann. Inst. H. Poincaré Anal. NonLinéaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004.

[20]

T. Lin and J. Wei, Multiple bound states of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 743-767.

[21]

Z. Liu and Z. Qang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.

[22]

C. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE. J. Quantum Electron., 23 (1987), 174-176. doi: 10.1109/JQE.1987.1073308.

[23]

S. Peng and Z. Wang, Segregated and Synchronized Vector Solutions for Nonlinear Schr\"odinger Systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0.

[24]

S. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Soviet Math. Dokl., 6 (1965), 1408-1411.

[25]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.

[26]

C. Swanson, The best Sobolev constant, Applicable Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.

[27]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pure Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[28]

J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei mat. Appl., 18 (2007), 279-293. doi: 10.4171/RLM/495.

[29]

J. Wei and T. Weth, Asymptotic behavior of solutions of planar systems with strong competition, Nonlinearity, 21 (2008), 305-317. doi: 10.1088/0951-7715/21/2/006.

[30]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N2$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1.

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