# American Institute of Mathematical Sciences

March  2013, 12(2): 695-710. doi: 10.3934/cpaa.2013.12.695

## Quasilinear systems involving multiple critical exponents and potentials

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

Received  July 2011 Revised  April 2012 Published  September 2012

In this paper, a quasilinear system of elliptic equations is investigated, which involves multiple critical Hardy--Sobolev exponents and Hardy--type terms. By variational methods and analytic technics, the existence of nontrivial solutions to the system is established.
Citation: Dongsheng Kang. Quasilinear systems involving multiple critical exponents and potentials. Communications on Pure and Applied Analysis, 2013, 12 (2) : 695-710. doi: 10.3934/cpaa.2013.12.695
##### References:
 [1] B. Abdellaoui, V. Felli and I. Peral, Existence and nonexistence for quasilinear equations involving the$p$-laplacian, Boll. Unione Mat. Ital. Sez., B8 (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] G. Arioli and F. Gazzola, Some results on $p-$Laplace equations with a critical growth term, Differential Integral Equations, 11 (1998), 311-326. [4] 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. [5] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259-275. [6] D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521-537. doi: 10.1016/j.jde.2004.03.005. [7] F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence(and nonexistence), and symmetry of extermal functions, Comm. Pure Appl. Math., 54 (2001), 229-257. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. [8] R. Dautray and P. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology, Physical Origins and Classical Methods," Vol. 1, Springer, Berlin, 1990. [9] A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177 (2001), 494-522. doi: 10.1006/jdeq.2000.3999. [10] D. Figueiredo, I. Peral and J. Rossi, The critical hyperbola for a Hamiltonian elliptic system with weights, Ann. Mat. Pura Appl., 187 (2008), doi: 10.1007/s10231-007-0054-1. [11] 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. [12] 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. [13] G. Hardy, J. Littlewood and G. Polya, "Inequalities," reprint of the 1952 edition, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1988. [14] 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.2009.02.024. [15] D. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms, Nonlinear Anal., 68 (2008), 1973-1985. doi: 10.1016/j.na.2007.01.024. [16] D. Kang, On the quasilinear elliptic problem with a critical Hardy-Sobolev exponent and a Hardy term, Nonlinear Anal., 69 (2008), 2432-2444. doi: 10.1016/j.na.2007.08.022. [17] D. Kang, Some properties of solutions to the singular quasilinear problem, Nonlinear Anal., 72 (2010), 682-688. doi: 10.1016/j.na.2009.07.009. [18] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), Rev. Mat. Iberoamericana, 1 (1985), 145-201. [19] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (II), Rev. Mat. Iberoamericana, 1 (1985), 45-121. [20] 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. [21] P. Rabinowitz, "Minimax Methods in Critical Points Theory with Applications to Defferential Equations," CBMS Regional Conference Series in Mathematics, Vol. 65, Providence, RI, 1986. [22] K. Sandeep, On the first eigenfunction of a perturbed Hardy-Sobolev operator, Nonlinear Differential Equations Appl., 10 (2003), 223-253. doi: 10.1007/s00030-003-1039-4. [23] K. Sreenadh, On the eigenvalue problem for the Hardy-Sobolev operator with indefinite weights, Elctronic J. Differential Equations, 2002 (2002), 1-12. [24] S. Terracini, On positive solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264. [25] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization, 12 (1984), 191-202. doi: 10.1007/BF01449041. [26] B. Xuan and J. Wang, Extremal functions and best constants to an inequality involving Hardy potential and critical Sobolev exponent, Nonlinear Anal., 71 (2009), 845-859. doi: 10.1016/j.na.2008.10.114.

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##### References:
 [1] B. Abdellaoui, V. Felli and I. Peral, Existence and nonexistence for quasilinear equations involving the$p$-laplacian, Boll. Unione Mat. Ital. Sez., B8 (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] G. Arioli and F. Gazzola, Some results on $p-$Laplace equations with a critical growth term, Differential Integral Equations, 11 (1998), 311-326. [4] 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. [5] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259-275. [6] D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521-537. doi: 10.1016/j.jde.2004.03.005. [7] F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence(and nonexistence), and symmetry of extermal functions, Comm. Pure Appl. Math., 54 (2001), 229-257. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. [8] R. Dautray and P. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology, Physical Origins and Classical Methods," Vol. 1, Springer, Berlin, 1990. [9] A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177 (2001), 494-522. doi: 10.1006/jdeq.2000.3999. [10] D. Figueiredo, I. Peral and J. Rossi, The critical hyperbola for a Hamiltonian elliptic system with weights, Ann. Mat. Pura Appl., 187 (2008), doi: 10.1007/s10231-007-0054-1. [11] 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. [12] 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. [13] G. Hardy, J. Littlewood and G. Polya, "Inequalities," reprint of the 1952 edition, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1988. [14] 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.2009.02.024. [15] D. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms, Nonlinear Anal., 68 (2008), 1973-1985. doi: 10.1016/j.na.2007.01.024. [16] D. Kang, On the quasilinear elliptic problem with a critical Hardy-Sobolev exponent and a Hardy term, Nonlinear Anal., 69 (2008), 2432-2444. doi: 10.1016/j.na.2007.08.022. [17] D. Kang, Some properties of solutions to the singular quasilinear problem, Nonlinear Anal., 72 (2010), 682-688. doi: 10.1016/j.na.2009.07.009. [18] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), Rev. Mat. Iberoamericana, 1 (1985), 145-201. [19] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (II), Rev. Mat. Iberoamericana, 1 (1985), 45-121. [20] 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. [21] P. Rabinowitz, "Minimax Methods in Critical Points Theory with Applications to Defferential Equations," CBMS Regional Conference Series in Mathematics, Vol. 65, Providence, RI, 1986. [22] K. Sandeep, On the first eigenfunction of a perturbed Hardy-Sobolev operator, Nonlinear Differential Equations Appl., 10 (2003), 223-253. doi: 10.1007/s00030-003-1039-4. [23] K. Sreenadh, On the eigenvalue problem for the Hardy-Sobolev operator with indefinite weights, Elctronic J. Differential Equations, 2002 (2002), 1-12. [24] S. Terracini, On positive solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264. [25] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization, 12 (1984), 191-202. doi: 10.1007/BF01449041. [26] B. Xuan and J. Wang, Extremal functions and best constants to an inequality involving Hardy potential and critical Sobolev exponent, Nonlinear Anal., 71 (2009), 845-859. doi: 10.1016/j.na.2008.10.114.
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