On elliptic systems with Sobolev critical exponent

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

On elliptic systems with Sobolev critical exponent

Yanfang Peng ^1,

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

Received October 2014 Revised October 2015 Published December 2015

Abstract
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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.

Keywords: Positive solutions, uniqueness, non-degeneracy, system, least energy solutions..

Mathematics Subject Classification: Primary: 35J20, 35J47; Secondary: 35J6.

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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