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

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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

John R. Graef ^1, , Shapour Heidarkhani ^2, and Lingju Kong ^1,

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

Abstract
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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).

Keywords: p_{n})$-biharmonic systems, critical points, Three solutions, $(p_{1}, variational methods, multiplicity results., \ldots.

Mathematics Subject Classification: Primary: 35J65, 47J1.

Citation: John R. Graef, Shapour Heidarkhani, Lingju Kong. Multiple solutions for a class of $(p_1, \ldots, p_n)$-biharmonic systems. Communications on Pure and 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), 135-143. doi: 10.1016/j.na.2007.11.038.
[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), 2594-2602. doi: 10.1016/j.na.2010.06.038.
[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., 15 (2011), 201-210.
[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), 1235-1250. doi: 10.4134/BKMS.2010.47.6.1235.
[5]	C. Amrouche, Singular boundary conditions and regularity for the biharmonic problem in the half-space, Commun. Pure Appl. Anal. 6 (2007), 957-982. doi: 10.3934/cpaa.2007.6.957.
[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), 93-103.
[7]	D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math., 39 (2009), 707-727. doi: 10.1216/RMJ-2009-39-3-707.
[8]	M. B. Ayed and M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six, Nonlinear Anal., 64 (2006), 924-957. doi: 10.1016/j.na.2005.05.050.
[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), 1705-1722.
[10]	Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357-368. doi: 10.1016/S0022-247X(02)00071-9.
[11]	L. Boccardo and D. Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differential Equations Appl., 9 (2002), 309-323. doi: 10.1007/s00030-002-8130-0.
[12]	G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176. doi: 10.1016/j.jmaa.2008.01.049.
[13]	Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239-267. doi: 10.1016/S0022-0396(02)00112-2.
[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), 1599-1612. doi: 10.1016/j.na.2006.08.002.
[15]	J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$, Nonlinear Anal., 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.
[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), 1033-1050. doi: 10.3934/dcds.2010.28.1033.
[17]	A. Djellit and S. Tas, On some nonlinear elliptic systems, Nonlinear Anal., 59 (2004), 695-706.
[18]	A. Djellit and S. Tas, Quasilinear elliptic systems with critical Sobolev exponents in $R^N$, Nonlinear Anal., 66 (2007), 1485-1497. doi: 10.1016/j.na.2006.02.005.
[19]	P. Drábek, N. M. Stavrakakis and N. B. Zographopoulos, Multiple nonsemitrivial solutions for quasilinear elliptic systems, Differential Integral Equations, 16 (2003), 1519-1531.
[20]	S. Federica, A biharmonic equation in $R^4$ involving nonlinearities with critical exponential growth, Commun. Pure Appl. Anal., 12 (2013), 405-428.
[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), 439-444. doi: 10.1016/j.aml.2004.03.011.
[22]	G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equation, Nonlinear Anal., 68 (2008), 3646-3656. doi: 10.1016/j.na.2007.04.007.
[23]	S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlinear Anal., 73 (2010), 547-554. doi: 10.1016/j.na.2010.03.051.
[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), 465-477. doi: 10.1017/S0013091504000112.
[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), 537-578. doi: 10.1137/1032120.
[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), 3322-3329. doi: 10.1016/j.na.2007.09.021.
[27]	C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal., 72 (2010), 1339-1347. doi: 10.1016/j.na.2009.08.011.
[28]	L. Li and C.-L. Tang, Existence of three solutions for (p,q)-biharmonic systems, Nonlinear Anal., 73 (2010), 796-805. doi: 10.1016/j.na.2010.04.018.
[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), 362-375. doi: 10.1016/j.jmaa.2006.04.021.
[30]	S. Liu and M. Squassina, On the existence of solutions to a fourth-order quasilinear resonant problem, Abstr. Appl. Anal., 7 (2002), 125-133. doi: 10.1155/S1085337502000805.
[31]	A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908. doi: 10.1016/S0362-546X(97)00446-X.
[32]	S. I. Pokhozhaev, On a constructive method of the calculus of variations, (Russian) Dokl. Akad. Nauk SSSR, 298 (1988), 1330-1333; translation in Soviet Math. Dokl., 37 (1988), 274-277.
[33]	B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.
[34]	B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084-3089. doi: 10.1016/j.na.2008.04.010.
[35]	J. Simon, Regularitè de la solution d'une equation non lineaire dans $R^N$, in "Journes d'Analyse Non Linaire" (P. Bénilan, J. Robert, eds.), Proc. Conf., Besanon, 1977, Lecture Notes in Math., 665, pp. 205-227, Springer, Berlin-Heidelberg-New York, 1978.
[36]	J. Su and Z. Liu, A bounded resonance problem for semilinear elliptic equations, Discrete Contin. Dyn. Syst., 19 (2007), 431-445. doi: 10.3934/dcds.2007.19.431.
[37]	T. Teramoto, On Positive radial entire solutions of second-order quasilinear elliptic systems, J. Math. Anal. Appl., 282 (2003), 531-552. doi: 10.1016/S0022-247X(03)00153-7.
[38]	W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738. doi: 10.1016/j.jmaa.2008.07.068.
[39]	Z. Wang, Nonradial positive solutions for a biharmonic critical growth problem, Commun. Pure Appl. Anal., 11 (2012), 517-545. doi: 10.3934/cpaa.2012.11.517.
[40]	G. Warnault, Regularity of the extremal solution for a biharmonic problem with general nonlinearity, Commun. Pure Appl. Anal., 8 (2009), 1709-1723. doi: 10.3934/cpaa.2009.8.1709.
[41]	E. Zeidler, "Nonlinear Functional Analysis and its Applications," Vol. II, Berlin-Heidelberg-New York, 1985.
[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), 1-10.
[43]	J. Zhang and S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60 (2005), 221-230.

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), 135-143. doi: 10.1016/j.na.2007.11.038.
[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), 2594-2602. doi: 10.1016/j.na.2010.06.038.
[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., 15 (2011), 201-210.
[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), 1235-1250. doi: 10.4134/BKMS.2010.47.6.1235.
[5]	C. Amrouche, Singular boundary conditions and regularity for the biharmonic problem in the half-space, Commun. Pure Appl. Anal. 6 (2007), 957-982. doi: 10.3934/cpaa.2007.6.957.
[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), 93-103.
[7]	D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math., 39 (2009), 707-727. doi: 10.1216/RMJ-2009-39-3-707.
[8]	M. B. Ayed and M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six, Nonlinear Anal., 64 (2006), 924-957. doi: 10.1016/j.na.2005.05.050.
[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), 1705-1722.
[10]	Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357-368. doi: 10.1016/S0022-247X(02)00071-9.
[11]	L. Boccardo and D. Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differential Equations Appl., 9 (2002), 309-323. doi: 10.1007/s00030-002-8130-0.
[12]	G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176. doi: 10.1016/j.jmaa.2008.01.049.
[13]	Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239-267. doi: 10.1016/S0022-0396(02)00112-2.
[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), 1599-1612. doi: 10.1016/j.na.2006.08.002.
[15]	J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$, Nonlinear Anal., 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.
[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), 1033-1050. doi: 10.3934/dcds.2010.28.1033.
[17]	A. Djellit and S. Tas, On some nonlinear elliptic systems, Nonlinear Anal., 59 (2004), 695-706.
[18]	A. Djellit and S. Tas, Quasilinear elliptic systems with critical Sobolev exponents in $R^N$, Nonlinear Anal., 66 (2007), 1485-1497. doi: 10.1016/j.na.2006.02.005.
[19]	P. Drábek, N. M. Stavrakakis and N. B. Zographopoulos, Multiple nonsemitrivial solutions for quasilinear elliptic systems, Differential Integral Equations, 16 (2003), 1519-1531.
[20]	S. Federica, A biharmonic equation in $R^4$ involving nonlinearities with critical exponential growth, Commun. Pure Appl. Anal., 12 (2013), 405-428.
[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), 439-444. doi: 10.1016/j.aml.2004.03.011.
[22]	G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equation, Nonlinear Anal., 68 (2008), 3646-3656. doi: 10.1016/j.na.2007.04.007.
[23]	S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlinear Anal., 73 (2010), 547-554. doi: 10.1016/j.na.2010.03.051.
[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), 465-477. doi: 10.1017/S0013091504000112.
[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), 537-578. doi: 10.1137/1032120.
[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), 3322-3329. doi: 10.1016/j.na.2007.09.021.
[27]	C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal., 72 (2010), 1339-1347. doi: 10.1016/j.na.2009.08.011.
[28]	L. Li and C.-L. Tang, Existence of three solutions for (p,q)-biharmonic systems, Nonlinear Anal., 73 (2010), 796-805. doi: 10.1016/j.na.2010.04.018.
[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), 362-375. doi: 10.1016/j.jmaa.2006.04.021.
[30]	S. Liu and M. Squassina, On the existence of solutions to a fourth-order quasilinear resonant problem, Abstr. Appl. Anal., 7 (2002), 125-133. doi: 10.1155/S1085337502000805.
[31]	A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908. doi: 10.1016/S0362-546X(97)00446-X.
[32]	S. I. Pokhozhaev, On a constructive method of the calculus of variations, (Russian) Dokl. Akad. Nauk SSSR, 298 (1988), 1330-1333; translation in Soviet Math. Dokl., 37 (1988), 274-277.
[33]	B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.
[34]	B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084-3089. doi: 10.1016/j.na.2008.04.010.
[35]	J. Simon, Regularitè de la solution d'une equation non lineaire dans $R^N$, in "Journes d'Analyse Non Linaire" (P. Bénilan, J. Robert, eds.), Proc. Conf., Besanon, 1977, Lecture Notes in Math., 665, pp. 205-227, Springer, Berlin-Heidelberg-New York, 1978.
[36]	J. Su and Z. Liu, A bounded resonance problem for semilinear elliptic equations, Discrete Contin. Dyn. Syst., 19 (2007), 431-445. doi: 10.3934/dcds.2007.19.431.
[37]	T. Teramoto, On Positive radial entire solutions of second-order quasilinear elliptic systems, J. Math. Anal. Appl., 282 (2003), 531-552. doi: 10.1016/S0022-247X(03)00153-7.
[38]	W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738. doi: 10.1016/j.jmaa.2008.07.068.
[39]	Z. Wang, Nonradial positive solutions for a biharmonic critical growth problem, Commun. Pure Appl. Anal., 11 (2012), 517-545. doi: 10.3934/cpaa.2012.11.517.
[40]	G. Warnault, Regularity of the extremal solution for a biharmonic problem with general nonlinearity, Commun. Pure Appl. Anal., 8 (2009), 1709-1723. doi: 10.3934/cpaa.2009.8.1709.
[41]	E. Zeidler, "Nonlinear Functional Analysis and its Applications," Vol. II, Berlin-Heidelberg-New York, 1985.
[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), 1-10.
[43]	J. Zhang and S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60 (2005), 221-230.

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