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. 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

show all references

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

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[2]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[3]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[4]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[5]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[6]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[7]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[8]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[9]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[10]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[11]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[12]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[13]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[14]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[15]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[16]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[17]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[18]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[19]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[20]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]