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December  2012, 32(12): 4247-4263. doi: 10.3934/dcds.2012.32.4247

Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$

1. 

School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China, China

Received  July 2011 Revised  December 2011 Published  August 2012

In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. By variational methods and analytic techniques, the best constant corresponding to the system is investigated, and the existence and nonexistence of ground state solutions to the system are established.
Citation: Dongsheng Kang, Fen Yang. Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4247-4263. doi: 10.3934/dcds.2012.32.4247
References:
[1]

B. Abdellaoui, V. Felli and I. Peral, Existence and nonexistence for quasilinear equations involving thep-laplacian, Boll. Unione Mat. Ital. Sez., B 8 (2006), 445-484.

[2]

B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137. doi: 10.1007/s00526-008-0177-2.

[3]

M. Bouchekif and Y. Nasri, On a singular elliptic system at resonance, Ann. Mat. Pura Appl., 189 (2010), 227-240. doi: 10.1007/s10231-009-0106-9.

[4]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372. doi: 10.1016/j.jde.2005.07.010.

[5]

V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. doi: 10.1080/03605300500394439.

[6]

D. Figueiredo, I. Peral and J. Rossi, The critical hyperbola for a Hamiltonian elliptic system with weights, Ann. Mat. Pura Appl., 187 (2008), 531-545. doi: 10.1007/s10231-007-0054-1.

[7]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5.

[8]

P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal., 61 (2005), 735-758. doi: 10.1016/j.na.2005.01.030.

[9]

G. Hardy, J. Littlewood and G. Polya, "Inequalities," 2nd, Cambridge University Press, Cambridge, 1988.

[10]

Y. Huang and D. Kang, Elliptic systems involving the critical exponents and potentials, Nonlinear Anal., 71 (2009), 3638-3653. doi: 10.1016/j.na.2009.02.024.

[11]

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.

[12]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426. doi: 10.1006/jdeq.1998.3589.

[13]

P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), Rev. Mat. Iberoamericana, (1) (1985), 145-201.

[14]

P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (II), Rev. Mat. Iberoamericana, 1(2) (1985), 45-121.

[15]

Z. Liu and P. Han, Existence of solutions for singular elliptic systems with critical exponents, Nonlinear Anal., 69 (2008), 2968-2983. doi: 10.1016/j.na.2007.08.073.

[16]

S. Terracini, On positive solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264.

[17]

J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization, 12 (1984), 191-202. doi: 10.1007/BF01449041.

[18]

M. Willem, "Analyse Fonctionnelle Élémentaire," Cassini Éditeurs, Paris, 2003.

show all references

References:
[1]

B. Abdellaoui, V. Felli and I. Peral, Existence and nonexistence for quasilinear equations involving thep-laplacian, Boll. Unione Mat. Ital. Sez., B 8 (2006), 445-484.

[2]

B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137. doi: 10.1007/s00526-008-0177-2.

[3]

M. Bouchekif and Y. Nasri, On a singular elliptic system at resonance, Ann. Mat. Pura Appl., 189 (2010), 227-240. doi: 10.1007/s10231-009-0106-9.

[4]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372. doi: 10.1016/j.jde.2005.07.010.

[5]

V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. doi: 10.1080/03605300500394439.

[6]

D. Figueiredo, I. Peral and J. Rossi, The critical hyperbola for a Hamiltonian elliptic system with weights, Ann. Mat. Pura Appl., 187 (2008), 531-545. doi: 10.1007/s10231-007-0054-1.

[7]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5.

[8]

P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal., 61 (2005), 735-758. doi: 10.1016/j.na.2005.01.030.

[9]

G. Hardy, J. Littlewood and G. Polya, "Inequalities," 2nd, Cambridge University Press, Cambridge, 1988.

[10]

Y. Huang and D. Kang, Elliptic systems involving the critical exponents and potentials, Nonlinear Anal., 71 (2009), 3638-3653. doi: 10.1016/j.na.2009.02.024.

[11]

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.

[12]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426. doi: 10.1006/jdeq.1998.3589.

[13]

P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), Rev. Mat. Iberoamericana, (1) (1985), 145-201.

[14]

P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (II), Rev. Mat. Iberoamericana, 1(2) (1985), 45-121.

[15]

Z. Liu and P. Han, Existence of solutions for singular elliptic systems with critical exponents, Nonlinear Anal., 69 (2008), 2968-2983. doi: 10.1016/j.na.2007.08.073.

[16]

S. Terracini, On positive solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264.

[17]

J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization, 12 (1984), 191-202. doi: 10.1007/BF01449041.

[18]

M. Willem, "Analyse Fonctionnelle Élémentaire," Cassini Éditeurs, Paris, 2003.

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