August  2012, 5(4): 789-796. doi: 10.3934/dcdss.2012.5.789

Multiple solutions for a perturbed system on strip-like domains

1. 

Department of Economics, Babeş-Bolyai University, Cluj-Napoca, str. Teodor Mihali, nr. 58-60, 400591, Romania

2. 

Department of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, str. M. Kogâlniceanu, nr. 1, 400584, Romania

Received  February 2011 Revised  March 2011 Published  November 2011

We prove a multiplicity result for a perturbed gradient-type system defined on strip-like domains. The approach is based on a recent Ricceri-type three critical point theorem.
Citation: Alexandru Kristály, Ildikó-Ilona Mezei. Multiple solutions for a perturbed system on strip-like domains. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 789-796. doi: 10.3934/dcdss.2012.5.789
References:
[1]

G. A. Afrouzi and S. Heidarkhani, Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the $(p_1,...,p_n)$-Laplacian,, Nonlin. Anal., 73 (2010), 2594. doi: 10.1016/j.na.2010.06.038. Google Scholar

[2]

G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $(p_1,...,p_n)$-Laplacian,, Nonlin. Anal., 70 (2009), 135. doi: 10.1016/j.na.2007.11.038. Google Scholar

[3]

L. Boccardo and G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations,, NoDEA Nonlinear Diff. Equ. Appl., 9 (2002), 309. Google Scholar

[4]

P. C. Carrião and O. H. Miyagaki, Existence of non-trivial solutions of elliptic variational systems in unbounded domains,, Nonlin. Anal., 51 (2002), 155. doi: 10.1016/S0362-546X(01)00817-3. Google Scholar

[5]

S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters,, Nonlin. Anal., 73 (2010), 547. doi: 10.1016/j.na.2010.03.051. Google Scholar

[6]

A. Kristály, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains,, Proc. Edinburgh Math. Soc. (2), 48 (2005), 465. doi: 10.1017/S0013091504000112. Google Scholar

[7]

C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian,, Nonlin. Anal., 69 (2008), 3322. doi: 10.1016/j.na.2007.09.021. Google Scholar

[8]

P.-L. Lions, Symétrie et compactité dans les espaces Sobolev,, J. Funct. Analysis, 49 (1982), 315. doi: 10.1016/0022-1236(82)90072-6. Google Scholar

[9]

R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. doi: 10.1007/BF01941322. Google Scholar

[10]

B. Ricceri, A further three critical points theorem,, Nonlin. Anal., 71 (2009), 4151. doi: 10.1016/j.na.2009.02.074. Google Scholar

show all references

References:
[1]

G. A. Afrouzi and S. Heidarkhani, Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the $(p_1,...,p_n)$-Laplacian,, Nonlin. Anal., 73 (2010), 2594. doi: 10.1016/j.na.2010.06.038. Google Scholar

[2]

G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $(p_1,...,p_n)$-Laplacian,, Nonlin. Anal., 70 (2009), 135. doi: 10.1016/j.na.2007.11.038. Google Scholar

[3]

L. Boccardo and G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations,, NoDEA Nonlinear Diff. Equ. Appl., 9 (2002), 309. Google Scholar

[4]

P. C. Carrião and O. H. Miyagaki, Existence of non-trivial solutions of elliptic variational systems in unbounded domains,, Nonlin. Anal., 51 (2002), 155. doi: 10.1016/S0362-546X(01)00817-3. Google Scholar

[5]

S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters,, Nonlin. Anal., 73 (2010), 547. doi: 10.1016/j.na.2010.03.051. Google Scholar

[6]

A. Kristály, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains,, Proc. Edinburgh Math. Soc. (2), 48 (2005), 465. doi: 10.1017/S0013091504000112. Google Scholar

[7]

C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian,, Nonlin. Anal., 69 (2008), 3322. doi: 10.1016/j.na.2007.09.021. Google Scholar

[8]

P.-L. Lions, Symétrie et compactité dans les espaces Sobolev,, J. Funct. Analysis, 49 (1982), 315. doi: 10.1016/0022-1236(82)90072-6. Google Scholar

[9]

R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. doi: 10.1007/BF01941322. Google Scholar

[10]

B. Ricceri, A further three critical points theorem,, Nonlin. Anal., 71 (2009), 4151. doi: 10.1016/j.na.2009.02.074. Google Scholar

[1]

Takahiro Hashimoto. Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains. Conference Publications, 2005, 2005 (Special) : 376-385. doi: 10.3934/proc.2005.2005.376

[2]

Takahiro Hashimoto, Mitsuharu Ôtani. Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 565-578. doi: 10.3934/dcds.1997.3.565

[3]

Luca Bisconti, Davide Catania. Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1861-1881. doi: 10.3934/cpaa.2017090

[4]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264

[5]

Eunkyoung Ko, Eun Kyoung Lee, R. Shivaji. Multiplicity results for classes of singular problems on an exterior domain. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5153-5166. doi: 10.3934/dcds.2013.33.5153

[6]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[7]

Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046

[8]

Roberta Filippucci, Chiara Lini. Existence of solutions for quasilinear Dirichlet problems with gradient terms. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 267-286. doi: 10.3934/dcdss.2019019

[9]

V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845

[10]

Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 947-973. doi: 10.3934/dcds.2000.6.947

[11]

Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073

[12]

Andrea Cianchi, Vladimir Maz'ya. Global gradient estimates in elliptic problems under minimal data and domain regularity. Communications on Pure & Applied Analysis, 2015, 14 (1) : 285-311. doi: 10.3934/cpaa.2015.14.285

[13]

E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 731-761. doi: 10.3934/dcds.2009.24.731

[14]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393

[15]

Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011

[16]

Yigui Ou, Haichan Lin. A class of accelerated conjugate-gradient-like methods based on a modified secant equation. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019013

[17]

Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-27. doi: 10.3934/dcdss.2020076

[18]

Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-24. doi: 10.3934/dcdss.2020092

[19]

Tommaso Leonori, Martina Magliocca. Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2923-2960. doi: 10.3934/cpaa.2019131

[20]

J. K. Krottje. On the dynamics of a mixed parabolic-gradient system. Communications on Pure & Applied Analysis, 2003, 2 (4) : 521-537. doi: 10.3934/cpaa.2003.2.521

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]