# American Institute of Mathematical Sciences

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

 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

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.
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.  Google Scholar [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.  Google Scholar [3] G. Bonanno and A. Chinnì, Discontinuous elliptic problems involving the $p(x)$-Laplacian,, Math. Nachr., 284 (2011), 639.  doi: 10.1002/mana.200810232.  Google Scholar [4] G. Bonanno and A. Chinnì, Multiple solutions for elliptic problems involving the $p(x)$-Laplacian,, Le Matematiche, LXVI (2011), 105.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition,, Classics Appl. Math., 5 (1990).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$,, Czechoslovak Math., 41 (1991), 592.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [24] B. Ricceri, On a three critical points theorem,, Arch. Math. (Basel), 75 (2000), 220.   Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] G. Bonanno and A. Chinnì, Discontinuous elliptic problems involving the $p(x)$-Laplacian,, Math. Nachr., 284 (2011), 639.  doi: 10.1002/mana.200810232.  Google Scholar [4] G. Bonanno and A. Chinnì, Multiple solutions for elliptic problems involving the $p(x)$-Laplacian,, Le Matematiche, LXVI (2011), 105.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition,, Classics Appl. Math., 5 (1990).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$,, Czechoslovak Math., 41 (1991), 592.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [24] B. Ricceri, On a three critical points theorem,, Arch. Math. (Basel), 75 (2000), 220.   Google Scholar [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.  Google Scholar
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