# American Institute of Mathematical Sciences

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 and 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), 2594-2602. doi: 10.1016/j.na.2010.06.038. [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), 135-143. doi: 10.1016/j.na.2007.11.038. [3] L. Boccardo and G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Diff. Equ. Appl., 9 (2002), 309-323. [4] P. C. Carrião and O. H. Miyagaki, Existence of non-trivial solutions of elliptic variational systems in unbounded domains, Nonlin. Anal., 51 (2002), 155-169. doi: 10.1016/S0362-546X(01)00817-3. [5] S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlin. Anal., 73 (2010), 547-554. doi: 10.1016/j.na.2010.03.051. [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), 465-477. doi: 10.1017/S0013091504000112. [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), 3322-3329. doi: 10.1016/j.na.2007.09.021. [8] P.-L. Lions, Symétrie et compactité dans les espaces Sobolev, J. Funct. Analysis, 49 (1982), 315-334. doi: 10.1016/0022-1236(82)90072-6. [9] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322. [10] B. Ricceri, A further three critical points theorem, Nonlin. Anal., 71 (2009), 4151-4157. doi: 10.1016/j.na.2009.02.074.

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##### 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), 2594-2602. doi: 10.1016/j.na.2010.06.038. [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), 135-143. doi: 10.1016/j.na.2007.11.038. [3] L. Boccardo and G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Diff. Equ. Appl., 9 (2002), 309-323. [4] P. C. Carrião and O. H. Miyagaki, Existence of non-trivial solutions of elliptic variational systems in unbounded domains, Nonlin. Anal., 51 (2002), 155-169. doi: 10.1016/S0362-546X(01)00817-3. [5] S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlin. Anal., 73 (2010), 547-554. doi: 10.1016/j.na.2010.03.051. [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), 465-477. doi: 10.1017/S0013091504000112. [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), 3322-3329. doi: 10.1016/j.na.2007.09.021. [8] P.-L. Lions, Symétrie et compactité dans les espaces Sobolev, J. Funct. Analysis, 49 (1982), 315-334. doi: 10.1016/0022-1236(82)90072-6. [9] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322. [10] B. Ricceri, A further three critical points theorem, Nonlin. Anal., 71 (2009), 4151-4157. doi: 10.1016/j.na.2009.02.074.
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