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 $\R^N$, 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.

show all references

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 $\R^N$, 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.

[1]

Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017

[2]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183

[3]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645

[4]

Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012

[5]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71

[6]

Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial and Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733

[7]

Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial and Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045

[8]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021037

[9]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143

[10]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

[11]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial and Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[12]

Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004

[13]

Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091

[14]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977

[15]

Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021095

[16]

Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021046

[17]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[18]

Jian-Wen Peng, Xin-Min Yang. Levitin-Polyak well-posedness of a system of generalized vector variational inequality problems. Journal of Industrial and Management Optimization, 2015, 11 (3) : 701-714. doi: 10.3934/jimo.2015.11.701

[19]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[20]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]