# American Institute of Mathematical Sciences

May  2013, 12(3): 1393-1406. doi: 10.3934/cpaa.2013.12.1393

## Multiple solutions for a class of $(p_1, \ldots, p_n)$-biharmonic systems

 1 Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States 2 Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran

Received  April 2012 Revised  July 2012 Published  September 2012

In this paper, the authors prove the existence of at least three weak solutions for the $(p_{1},\ldots,p_{n})$-- biharmonic system $$\begin{cases} \Delta(|\Delta u_{i}|^{p_i-2}\Delta u_{i}) = \lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}), & \mbox{in} \ \Omega,\\ u_{i}=\Delta u_i=0, & \mbox{on} \ \partial\Omega. \end{cases}$$ The main tool is a recent three critical points theorem of Averna and Bonanno ({\it A three critical points theorem and its applications to the ordinary Dirichlet problem}, Topol. Methods Nonlinear Anal. 22 (2003), 93-104).
Citation: John R. Graef, Shapour Heidarkhani, Lingju Kong. Multiple solutions for a class of $(p_1, \ldots, p_n)$-biharmonic systems. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1393-1406. doi: 10.3934/cpaa.2013.12.1393
##### References:
 [1] G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian,, Nonlinear Anal., 70 (2009), 135. doi: 10.1016/j.na.2007.11.038. Google Scholar [2] G. A. Afrouzi and S. Heidarkhani, Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian, , Nonlinear Anal., 73 (2010), 2594. doi: 10.1016/j.na.2010.06.038. Google Scholar [3] G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem,, Taiwanese J. Math., (2011), 201. Google Scholar [4] G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a $(p_1, \ldots, p_n)$-Laplacian, , Bull. Korean Math. Soc., 47 (2010), 1235. doi: 10.4134/BKMS.2010.47.6.1235. Google Scholar [5] C. Amrouche, Singular boundary conditions and regularity for the biharmonic problem in the half-space,, Commun. Pure Appl. Anal. 6 (2007), 6 (2007), 957. doi: 10.3934/cpaa.2007.6.957. Google Scholar [6] D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem,, Topol. Methods Nonlinear Anal., 22 (2003), 93. Google Scholar [7] D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions,, Rocky Mountain J. Math., 39 (2009), 707. doi: 10.1216/RMJ-2009-39-3-707. Google Scholar [8] M. B. Ayed and M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six,, Nonlinear Anal., 64 (2006), 924. doi: 10.1016/j.na.2005.05.050. Google Scholar [9] M. Ayed and A. Selmi, Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain,, Commun. Pure Appl. Anal., 9 (2012), 1705. Google Scholar [10] Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations,, J. Math. Anal. Appl., 270 (2002), 357. doi: 10.1016/S0022-247X(02)00071-9. Google Scholar [11] L. Boccardo and D. Figueiredo, Some remarks on a system of quasilinear elliptic equations,, Nonlinear Differential Equations Appl., 9 (2002), 309. doi: 10.1007/s00030-002-8130-0. Google Scholar [12] G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations,, J. Math. Anal. Appl., 343 (2008), 1166. doi: 10.1016/j.jmaa.2008.01.049. Google Scholar [13] Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method,, J. Differential Equations, 190 (2003), 239. doi: 10.1016/S0022-0396(02)00112-2. Google Scholar [14] A. Cabada, J. A. Cid and L. Sanchez, Positivity and lower and upper solutions for fourth-order boundary value problems,, Nonlinear Anal., 67 (2007), 1599. doi: 10.1016/j.na.2006.08.002. Google Scholar [15] J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$,, Nonlinear Anal., 49 (2002), 861. doi: 10.1016/S0362-546X(01)00144-4. Google Scholar [16] C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains,, Discrete Contin. Dyn. Syst., 28 (2010), 1033. doi: 10.3934/dcds.2010.28.1033. Google Scholar [17] A. Djellit and S. Tas, On some nonlinear elliptic systems,, Nonlinear Anal., 59 (2004), 695. Google Scholar [18] A. Djellit and S. Tas, Quasilinear elliptic systems with critical Sobolev exponents in $R^N$,, Nonlinear Anal., 66 (2007), 1485. doi: 10.1016/j.na.2006.02.005. Google Scholar [19] P. Drábek, N. M. Stavrakakis and N. B. Zographopoulos, Multiple nonsemitrivial solutions for quasilinear elliptic systems,, Differential Integral Equations, 16 (2003), 1519. Google Scholar [20] S. Federica, A biharmonic equation in $R^4$ involving nonlinearities with critical exponential growth,, Commun. Pure Appl. Anal., 12 (2013), 405. Google Scholar [21] M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation,, Appl. Math. Lett., 18 (2005), 439. doi: 10.1016/j.aml.2004.03.011. Google Scholar [22] G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equation,, Nonlinear Anal., 68 (2008), 3646. doi: 10.1016/j.na.2007.04.007. Google Scholar [23] S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters,, Nonlinear Anal., 73 (2010), 547. doi: 10.1016/j.na.2010.03.051. Google Scholar [24] A. Kristály, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains,, Proc. Edinb. Math. Soc., 48 (2005), 465. doi: 10.1017/S0013091504000112. Google Scholar [25] A. C. Lazer and P. J. McKenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, SIAM Rev., 32 (1990), 537. doi: 10.1137/1032120. Google Scholar [26] C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian,, Nonlinear Anal., 69 (2008), 3322. doi: 10.1016/j.na.2007.09.021. Google Scholar [27] C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic,, Nonlinear Anal., 72 (2010), 1339. doi: 10.1016/j.na.2009.08.011. Google Scholar [28] L. Li and C.-L. Tang, Existence of three solutions for (p,q)-biharmonic systems,, Nonlinear Anal., 73 (2010), 796. doi: 10.1016/j.na.2010.04.018. Google Scholar [29] X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters,, J. Math. Anal. Appl., 327 (2007), 362. doi: 10.1016/j.jmaa.2006.04.021. Google Scholar [30] S. Liu and M. Squassina, On the existence of solutions to a fourth-order quasilinear resonant problem,, Abstr. Appl. Anal., 7 (2002), 125. doi: 10.1155/S1085337502000805. Google Scholar [31] A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem,, Nonlinear Anal., 31 (1998), 895. doi: 10.1016/S0362-546X(97)00446-X. Google Scholar [32] S. I. Pokhozhaev, On a constructive method of the calculus of variations, (Russian), Dokl. Akad. Nauk SSSR, 298 (1988), 1330. Google Scholar [33] B. Ricceri, On a three critical points theorem,, Arch. Math. (Basel), 75 (2000), 220. Google Scholar [34] B. Ricceri, A three critical points theorem revisited,, Nonlinear Anal., 70 (2009), 3084. doi: 10.1016/j.na.2008.04.010. Google Scholar [35] J. Simon, Regularitè de la solution d'une equation non lineaire dans $R^N$,, in, (1977), 205. Google Scholar [36] J. Su and Z. Liu, A bounded resonance problem for semilinear elliptic equations,, Discrete Contin. Dyn. Syst., 19 (2007), 431. doi: 10.3934/dcds.2007.19.431. Google Scholar [37] T. Teramoto, On Positive radial entire solutions of second-order quasilinear elliptic systems,, J. Math. Anal. Appl., 282 (2003), 531. doi: 10.1016/S0022-247X(03)00153-7. Google Scholar [38] W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type,, J. Math. Anal. Appl., 348 (2008), 730. doi: 10.1016/j.jmaa.2008.07.068. Google Scholar [39] Z. Wang, Nonradial positive solutions for a biharmonic critical growth problem,, Commun. Pure Appl. Anal., 11 (2012), 517. doi: 10.3934/cpaa.2012.11.517. Google Scholar [40] G. Warnault, Regularity of the extremal solution for a biharmonic problem with general nonlinearity,, Commun. Pure Appl. Anal., 8 (2009), 1709. doi: 10.3934/cpaa.2009.8.1709. Google Scholar [41] E. Zeidler, "Nonlinear Functional Analysis and its Applications," Vol. II,, Berlin-Heidelberg-New York, (1985). Google Scholar [42] G. Q. Zhang, X. P. Liu and S. Y. Liu, Remarks on a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian,, Electron. J. Differential Equations, 20 (2005), 1. Google Scholar [43] J. Zhang and S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems,, Nonlinear Anal., 60 (2005), 221. Google Scholar

show all references

##### References:
 [1] G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian,, Nonlinear Anal., 70 (2009), 135. doi: 10.1016/j.na.2007.11.038. Google Scholar [2] G. A. Afrouzi and S. Heidarkhani, Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian, , Nonlinear Anal., 73 (2010), 2594. doi: 10.1016/j.na.2010.06.038. Google Scholar [3] G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem,, Taiwanese J. Math., (2011), 201. Google Scholar [4] G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a $(p_1, \ldots, p_n)$-Laplacian, , Bull. Korean Math. Soc., 47 (2010), 1235. doi: 10.4134/BKMS.2010.47.6.1235. Google Scholar [5] C. Amrouche, Singular boundary conditions and regularity for the biharmonic problem in the half-space,, Commun. Pure Appl. Anal. 6 (2007), 6 (2007), 957. doi: 10.3934/cpaa.2007.6.957. Google Scholar [6] D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem,, Topol. Methods Nonlinear Anal., 22 (2003), 93. Google Scholar [7] D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions,, Rocky Mountain J. Math., 39 (2009), 707. doi: 10.1216/RMJ-2009-39-3-707. Google Scholar [8] M. B. Ayed and M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six,, Nonlinear Anal., 64 (2006), 924. doi: 10.1016/j.na.2005.05.050. Google Scholar [9] M. Ayed and A. Selmi, Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain,, Commun. Pure Appl. Anal., 9 (2012), 1705. Google Scholar [10] Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations,, J. Math. Anal. Appl., 270 (2002), 357. doi: 10.1016/S0022-247X(02)00071-9. Google Scholar [11] L. Boccardo and D. Figueiredo, Some remarks on a system of quasilinear elliptic equations,, Nonlinear Differential Equations Appl., 9 (2002), 309. doi: 10.1007/s00030-002-8130-0. Google Scholar [12] G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations,, J. Math. Anal. Appl., 343 (2008), 1166. doi: 10.1016/j.jmaa.2008.01.049. Google Scholar [13] Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method,, J. Differential Equations, 190 (2003), 239. doi: 10.1016/S0022-0396(02)00112-2. Google Scholar [14] A. Cabada, J. A. Cid and L. Sanchez, Positivity and lower and upper solutions for fourth-order boundary value problems,, Nonlinear Anal., 67 (2007), 1599. doi: 10.1016/j.na.2006.08.002. Google Scholar [15] J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$,, Nonlinear Anal., 49 (2002), 861. doi: 10.1016/S0362-546X(01)00144-4. Google Scholar [16] C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains,, Discrete Contin. Dyn. Syst., 28 (2010), 1033. doi: 10.3934/dcds.2010.28.1033. Google Scholar [17] A. Djellit and S. Tas, On some nonlinear elliptic systems,, Nonlinear Anal., 59 (2004), 695. Google Scholar [18] A. Djellit and S. Tas, Quasilinear elliptic systems with critical Sobolev exponents in $R^N$,, Nonlinear Anal., 66 (2007), 1485. doi: 10.1016/j.na.2006.02.005. Google Scholar [19] P. Drábek, N. M. Stavrakakis and N. B. Zographopoulos, Multiple nonsemitrivial solutions for quasilinear elliptic systems,, Differential Integral Equations, 16 (2003), 1519. Google Scholar [20] S. Federica, A biharmonic equation in $R^4$ involving nonlinearities with critical exponential growth,, Commun. Pure Appl. Anal., 12 (2013), 405. Google Scholar [21] M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation,, Appl. Math. Lett., 18 (2005), 439. doi: 10.1016/j.aml.2004.03.011. Google Scholar [22] G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equation,, Nonlinear Anal., 68 (2008), 3646. doi: 10.1016/j.na.2007.04.007. Google Scholar [23] S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters,, Nonlinear Anal., 73 (2010), 547. doi: 10.1016/j.na.2010.03.051. Google Scholar [24] A. Kristály, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains,, Proc. Edinb. Math. Soc., 48 (2005), 465. doi: 10.1017/S0013091504000112. Google Scholar [25] A. C. Lazer and P. J. McKenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, SIAM Rev., 32 (1990), 537. doi: 10.1137/1032120. Google Scholar [26] C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian,, Nonlinear Anal., 69 (2008), 3322. doi: 10.1016/j.na.2007.09.021. Google Scholar [27] C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic,, Nonlinear Anal., 72 (2010), 1339. doi: 10.1016/j.na.2009.08.011. Google Scholar [28] L. Li and C.-L. Tang, Existence of three solutions for (p,q)-biharmonic systems,, Nonlinear Anal., 73 (2010), 796. doi: 10.1016/j.na.2010.04.018. Google Scholar [29] X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters,, J. Math. Anal. Appl., 327 (2007), 362. doi: 10.1016/j.jmaa.2006.04.021. Google Scholar [30] S. Liu and M. Squassina, On the existence of solutions to a fourth-order quasilinear resonant problem,, Abstr. Appl. Anal., 7 (2002), 125. doi: 10.1155/S1085337502000805. Google Scholar [31] A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem,, Nonlinear Anal., 31 (1998), 895. doi: 10.1016/S0362-546X(97)00446-X. Google Scholar [32] S. I. Pokhozhaev, On a constructive method of the calculus of variations, (Russian), Dokl. Akad. Nauk SSSR, 298 (1988), 1330. Google Scholar [33] B. Ricceri, On a three critical points theorem,, Arch. Math. (Basel), 75 (2000), 220. Google Scholar [34] B. Ricceri, A three critical points theorem revisited,, Nonlinear Anal., 70 (2009), 3084. doi: 10.1016/j.na.2008.04.010. Google Scholar [35] J. Simon, Regularitè de la solution d'une equation non lineaire dans $R^N$,, in, (1977), 205. Google Scholar [36] J. Su and Z. Liu, A bounded resonance problem for semilinear elliptic equations,, Discrete Contin. Dyn. Syst., 19 (2007), 431. doi: 10.3934/dcds.2007.19.431. Google Scholar [37] T. Teramoto, On Positive radial entire solutions of second-order quasilinear elliptic systems,, J. Math. Anal. Appl., 282 (2003), 531. doi: 10.1016/S0022-247X(03)00153-7. Google Scholar [38] W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type,, J. Math. Anal. Appl., 348 (2008), 730. doi: 10.1016/j.jmaa.2008.07.068. Google Scholar [39] Z. Wang, Nonradial positive solutions for a biharmonic critical growth problem,, Commun. Pure Appl. Anal., 11 (2012), 517. doi: 10.3934/cpaa.2012.11.517. Google Scholar [40] G. Warnault, Regularity of the extremal solution for a biharmonic problem with general nonlinearity,, Commun. Pure Appl. Anal., 8 (2009), 1709. doi: 10.3934/cpaa.2009.8.1709. Google Scholar [41] E. Zeidler, "Nonlinear Functional Analysis and its Applications," Vol. II,, Berlin-Heidelberg-New York, (1985). Google Scholar [42] G. Q. Zhang, X. P. Liu and S. Y. Liu, Remarks on a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian,, Electron. J. Differential Equations, 20 (2005), 1. Google Scholar [43] J. Zhang and S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems,, Nonlinear Anal., 60 (2005), 221. Google Scholar
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