Multiple solutions for a Neumann-type differential inclusion problem involving the p(\cdot)-Laplacian

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August 2012, 5(4): 753-764. doi: 10.3934/dcdss.2012.5.753

Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian

Antonia Chinnì ^1, and Roberto Livrea ^2,

1.	Department of Science for Engineering and Architecture (Mathematics Section), Engineering Faculty, University of Messina, Messina, 98166, Italy
2.	Department MECMAT, Engineering Faculty, University of Reggio Calabria, Reggio Calabria, 89100, Italy

Received April 2011 Revised August 2011 Published November 2011

Abstract
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Using a multiple critical points theorem for locally Lipschitz continuous functionals, we establish the existence of at least three distinct solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian.

Keywords: three-critical-points theorem., variable exponent Sobolev space, $p(x)$-Laplacian, critical points of locally Lipschitz continuous functionals, differential inclusion problem.

Mathematics Subject Classification: Primary: 35J20, 35R7.

Citation: Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753

References:

[1]	G. Bonanno, Some remarks on a three critical points theorem,, Nonlinear Anal., 54 (2003), 651. doi: 10.1016/S0362-546X(03)00092-0.
[2]	G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,, J. Differential Equations, 244 (2008), 3031. doi: 10.1016/j.jde.2008.02.025.
[3]	G. Bonanno and A. Chinnì, Discontinuous elliptic problems involving the $p(x)$-Laplacian,, Math. Nachr., 284 (2011), 639. doi: 10.1002/mana.200810232.
[4]	G. Bonanno and A. Chinnì, Multiple solutions for elliptic problems involving the $p(x)$-Laplacian,, Le Matematiche, LXVI (2011), 105.
[5]	G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition,, Appl. Anal., 89 (2010), 1. doi: 10.1080/00036810903397438.
[6]	F. Cammaroto, A. Chinnì and B. Di Bella, Multiple solutions for a Neumann problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 71 (2009), 4486. doi: 10.1016/j.na.2009.03.009.
[7]	K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations,, J. Math. Anal. Appl., 80 (1981), 102. doi: 10.1016/0022-247X(81)90095-0.
[8]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition,, Classics Appl. Math., 5 (1990).
[9]	G. Dai, Three solutions for a Neumann-type differential inclusion problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 70 (2009), 3755. doi: 10.1016/j.na.2008.07.031.
[10]	G. Dai, Infinitely many solutions for a Neumann-type differential inclusion problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 70 (2009), 2297. doi: 10.1016/j.na.2008.03.009.
[11]	X. Fan and S.-G. Deng, Remarks on Ricceri's variational principle and applications to the $p(x)$-Laplacian equations,, Nonlinear Anal., 67 (2007), 3064. doi: 10.1016/j.na.2006.09.060.
[12]	X. Fan and C. Ji, Existence of infinitely many solutions for a Neumann problem involving the $p(x)$-Laplacian,, J. Math. Anal. Appl., 334 (2007), 248. doi: 10.1016/j.jmaa.2006.12.055.
[13]	X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, J. Math. Anal. Appl., 263 (2001), 424. doi: 10.1006/jmaa.2000.7617.
[14]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$,, Czechoslovak Math., 41 (1991), 592.
[15]	A. Kristály, Infinitely many solutions for a differential inclusion problem in $\mathbbR^n$,, J. Differential Equations, 220 (2006), 511. doi: 10.1016/j.jde.2005.02.007.
[16]	A. Kristály, M. Mihǎilescu and V. Rǎdulescu, Two non-trivial solutions for a non-homogeneous Neumann problem: An Orlicz-Sobolev space setting,, Proc. Royal Soc. Edinburgh Sect. A, 139 (2009), 367. doi: 10.1017/S030821050700025X.
[17]	S. A. Marano and D. Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems,, Nonlinear Anal., 48 (2002), 37. doi: 10.1016/S0362-546X(00)00171-1.
[18]	S. A. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian,, J. Differential Equations, 182 (2002), 108. doi: 10.1006/jdeq.2001.4092.
[19]	M. Mihǎilescu, Existence and multiplicity of solutions for a Neumann problem involving the $p(x)$-Laplace operator,, Nonlinear Analysis, 67 (2007), 1419. doi: 10.1016/j.na.2006.07.027.
[20]	D. S. Moschetto, A quasilinear Neumann problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 71 (2009), 2739. doi: 10.1016/j.na.2009.01.109.
[21]	D. Motreanu and N. S. Papageorgiou, On some elliptic hemivariational and variational-hemivariational inequalities,, Nonlinear Anal., 62 (2005), 757. doi: 10.1016/j.na.2005.03.101.
[22]	N. S. Papageorgiou and E. M. Rocha, "Existence and Multiplicity of Solutions for the Noncoercive Neumann p-Laplacian,", Preceedings of the 2007 Conference on Variational and Toplogical Methods: Theory, 18 (2010), 57.
[23]	N. S. Papageorgiou and G. Smyrlis, Multiple solutions for nonlinear Neumann problems with the p-Laplacian and a nonsmooth crossing potential,, Nonlinearity, 23 (2010), 529. doi: 10.1088/0951-7715/23/3/005.
[24]	B. Ricceri, On a three critical points theorem,, Arch. Math. (Basel), 75 (2000), 220.
[25]	B. Ricceri, A general variational principle and some of its applications,, J. Comput. Appl. Math., 113 (2000). doi: 10.1016/S0377-0427(99)00269-1.

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References:

[1]	G. Bonanno, Some remarks on a three critical points theorem,, Nonlinear Anal., 54 (2003), 651. doi: 10.1016/S0362-546X(03)00092-0.
[2]	G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,, J. Differential Equations, 244 (2008), 3031. doi: 10.1016/j.jde.2008.02.025.
[3]	G. Bonanno and A. Chinnì, Discontinuous elliptic problems involving the $p(x)$-Laplacian,, Math. Nachr., 284 (2011), 639. doi: 10.1002/mana.200810232.
[4]	G. Bonanno and A. Chinnì, Multiple solutions for elliptic problems involving the $p(x)$-Laplacian,, Le Matematiche, LXVI (2011), 105.
[5]	G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition,, Appl. Anal., 89 (2010), 1. doi: 10.1080/00036810903397438.
[6]	F. Cammaroto, A. Chinnì and B. Di Bella, Multiple solutions for a Neumann problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 71 (2009), 4486. doi: 10.1016/j.na.2009.03.009.
[7]	K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations,, J. Math. Anal. Appl., 80 (1981), 102. doi: 10.1016/0022-247X(81)90095-0.
[8]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition,, Classics Appl. Math., 5 (1990).
[9]	G. Dai, Three solutions for a Neumann-type differential inclusion problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 70 (2009), 3755. doi: 10.1016/j.na.2008.07.031.
[10]	G. Dai, Infinitely many solutions for a Neumann-type differential inclusion problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 70 (2009), 2297. doi: 10.1016/j.na.2008.03.009.
[11]	X. Fan and S.-G. Deng, Remarks on Ricceri's variational principle and applications to the $p(x)$-Laplacian equations,, Nonlinear Anal., 67 (2007), 3064. doi: 10.1016/j.na.2006.09.060.
[12]	X. Fan and C. Ji, Existence of infinitely many solutions for a Neumann problem involving the $p(x)$-Laplacian,, J. Math. Anal. Appl., 334 (2007), 248. doi: 10.1016/j.jmaa.2006.12.055.
[13]	X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, J. Math. Anal. Appl., 263 (2001), 424. doi: 10.1006/jmaa.2000.7617.
[14]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$,, Czechoslovak Math., 41 (1991), 592.
[15]	A. Kristály, Infinitely many solutions for a differential inclusion problem in $\mathbbR^n$,, J. Differential Equations, 220 (2006), 511. doi: 10.1016/j.jde.2005.02.007.
[16]	A. Kristály, M. Mihǎilescu and V. Rǎdulescu, Two non-trivial solutions for a non-homogeneous Neumann problem: An Orlicz-Sobolev space setting,, Proc. Royal Soc. Edinburgh Sect. A, 139 (2009), 367. doi: 10.1017/S030821050700025X.
[17]	S. A. Marano and D. Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems,, Nonlinear Anal., 48 (2002), 37. doi: 10.1016/S0362-546X(00)00171-1.
[18]	S. A. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian,, J. Differential Equations, 182 (2002), 108. doi: 10.1006/jdeq.2001.4092.
[19]	M. Mihǎilescu, Existence and multiplicity of solutions for a Neumann problem involving the $p(x)$-Laplace operator,, Nonlinear Analysis, 67 (2007), 1419. doi: 10.1016/j.na.2006.07.027.
[20]	D. S. Moschetto, A quasilinear Neumann problem involving the $p(x)$-Laplacian,, Nonlinear Anal., 71 (2009), 2739. doi: 10.1016/j.na.2009.01.109.
[21]	D. Motreanu and N. S. Papageorgiou, On some elliptic hemivariational and variational-hemivariational inequalities,, Nonlinear Anal., 62 (2005), 757. doi: 10.1016/j.na.2005.03.101.
[22]	N. S. Papageorgiou and E. M. Rocha, "Existence and Multiplicity of Solutions for the Noncoercive Neumann p-Laplacian,", Preceedings of the 2007 Conference on Variational and Toplogical Methods: Theory, 18 (2010), 57.
[23]	N. S. Papageorgiou and G. Smyrlis, Multiple solutions for nonlinear Neumann problems with the p-Laplacian and a nonsmooth crossing potential,, Nonlinearity, 23 (2010), 529. doi: 10.1088/0951-7715/23/3/005.
[24]	B. Ricceri, On a three critical points theorem,, Arch. Math. (Basel), 75 (2000), 220.
[25]	B. Ricceri, A general variational principle and some of its applications,, J. Comput. Appl. Math., 113 (2000). doi: 10.1016/S0377-0427(99)00269-1.

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