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

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.   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

[1]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[2]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018

[3]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[4]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178

[5]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[6]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405

[7]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[8]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

[9]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285

[10]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[11]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354

[12]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[13]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[14]

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

[15]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[16]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061

[17]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455

[18]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[19]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298

[20]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]