# 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 & 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.   Google Scholar [2] B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\R^N$,, Calc. Var. Partial Differential Equations, 34 (2009), 97.  doi: 10.1007/s00526-008-0177-2.  Google Scholar [3] G. Arioli and F. Gazzola, Some results on $p-$Laplace equations with a critical growth term,, Differential Integral Equations, 11 (1998), 311.   Google Scholar [4] M. Bouchekif and Y. Nasri, On a singular elliptic system at resonance,, Ann. Mat. Pura Appl., 189 (2010), 227.  doi: 10.1007/s10231-009-0106-9.  Google Scholar [5] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights,, Compos. Math., 53 (1984), 259.   Google Scholar [6] D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential,, J. Differential Equations, 205 (2004), 521.  doi: 10.1016/j.jde.2004.03.005.  Google Scholar [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.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar [8] R. Dautray and P. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology, Physical Origins and Classical Methods,", Vol. 1, (1990).   Google Scholar [9] A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations,, J. Differential Equations, 177 (2001), 494.  doi: 10.1006/jdeq.2000.3999.  Google Scholar [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.  Google Scholar [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.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar [12] P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms,, Nonlinear Anal., 61 (2005), 735.  doi: 10.1016/j.na.2005.01.030.  Google Scholar [13] G. Hardy, J. Littlewood and G. Polya, "Inequalities,", reprint of the 1952 edition, (1952).   Google Scholar [14] 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.2009.02.024.  Google Scholar [15] D. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms,, Nonlinear Anal., 68 (2008), 1973.  doi: 10.1016/j.na.2007.01.024.  Google Scholar [16] D. Kang, On the quasilinear elliptic problem with a critical Hardy-Sobolev exponent and a Hardy term,, Nonlinear Anal., 69 (2008), 2432.  doi: 10.1016/j.na.2007.08.022.  Google Scholar [17] D. Kang, Some properties of solutions to the singular quasilinear problem,, Nonlinear Anal., 72 (2010), 682.  doi: 10.1016/j.na.2009.07.009.  Google Scholar [18] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), , Rev. Mat. Iberoamericana, 1 (1985), 145.   Google Scholar [19] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (II), , Rev. Mat. Iberoamericana, 1 (1985), 45.   Google Scholar [20] Z. Liu and P. Han, Existence of solutions for singular elliptic systems with critical exponents,, Nonlinear Anal., 69 (2008), 2968.  doi: 10.1016/j.na.2007.08.073.  Google Scholar [21] P. Rabinowitz, "Minimax Methods in Critical Points Theory with Applications to Defferential Equations,", CBMS Regional Conference Series in Mathematics, (1986).   Google Scholar [22] K. Sandeep, On the first eigenfunction of a perturbed Hardy-Sobolev operator,, Nonlinear Differential Equations Appl., 10 (2003), 223.  doi: 10.1007/s00030-003-1039-4.  Google Scholar [23] K. Sreenadh, On the eigenvalue problem for the Hardy-Sobolev operator with indefinite weights,, Elctronic J. Differential Equations, 2002 (2002), 1.   Google Scholar [24] S. Terracini, On positive solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Equations, 2 (1996), 241.   Google Scholar [25] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optimization, 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar [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.  doi: 10.1016/j.na.2008.10.114.  Google Scholar

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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.   Google Scholar [2] B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\R^N$,, Calc. Var. Partial Differential Equations, 34 (2009), 97.  doi: 10.1007/s00526-008-0177-2.  Google Scholar [3] G. Arioli and F. Gazzola, Some results on $p-$Laplace equations with a critical growth term,, Differential Integral Equations, 11 (1998), 311.   Google Scholar [4] M. Bouchekif and Y. Nasri, On a singular elliptic system at resonance,, Ann. Mat. Pura Appl., 189 (2010), 227.  doi: 10.1007/s10231-009-0106-9.  Google Scholar [5] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights,, Compos. Math., 53 (1984), 259.   Google Scholar [6] D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential,, J. Differential Equations, 205 (2004), 521.  doi: 10.1016/j.jde.2004.03.005.  Google Scholar [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.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar [8] R. Dautray and P. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology, Physical Origins and Classical Methods,", Vol. 1, (1990).   Google Scholar [9] A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations,, J. Differential Equations, 177 (2001), 494.  doi: 10.1006/jdeq.2000.3999.  Google Scholar [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.  Google Scholar [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.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar [12] P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms,, Nonlinear Anal., 61 (2005), 735.  doi: 10.1016/j.na.2005.01.030.  Google Scholar [13] G. Hardy, J. Littlewood and G. Polya, "Inequalities,", reprint of the 1952 edition, (1952).   Google Scholar [14] 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.2009.02.024.  Google Scholar [15] D. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms,, Nonlinear Anal., 68 (2008), 1973.  doi: 10.1016/j.na.2007.01.024.  Google Scholar [16] D. Kang, On the quasilinear elliptic problem with a critical Hardy-Sobolev exponent and a Hardy term,, Nonlinear Anal., 69 (2008), 2432.  doi: 10.1016/j.na.2007.08.022.  Google Scholar [17] D. Kang, Some properties of solutions to the singular quasilinear problem,, Nonlinear Anal., 72 (2010), 682.  doi: 10.1016/j.na.2009.07.009.  Google Scholar [18] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), , Rev. Mat. Iberoamericana, 1 (1985), 145.   Google Scholar [19] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (II), , Rev. Mat. Iberoamericana, 1 (1985), 45.   Google Scholar [20] Z. Liu and P. Han, Existence of solutions for singular elliptic systems with critical exponents,, Nonlinear Anal., 69 (2008), 2968.  doi: 10.1016/j.na.2007.08.073.  Google Scholar [21] P. Rabinowitz, "Minimax Methods in Critical Points Theory with Applications to Defferential Equations,", CBMS Regional Conference Series in Mathematics, (1986).   Google Scholar [22] K. Sandeep, On the first eigenfunction of a perturbed Hardy-Sobolev operator,, Nonlinear Differential Equations Appl., 10 (2003), 223.  doi: 10.1007/s00030-003-1039-4.  Google Scholar [23] K. Sreenadh, On the eigenvalue problem for the Hardy-Sobolev operator with indefinite weights,, Elctronic J. Differential Equations, 2002 (2002), 1.   Google Scholar [24] S. Terracini, On positive solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Equations, 2 (1996), 241.   Google Scholar [25] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optimization, 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar [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.  doi: 10.1016/j.na.2008.10.114.  Google Scholar
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