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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

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), 651-665. 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), 3031-3059. 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), 639-652. 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 - Fasc. I (2011), 105-113. 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), 1-10. 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), 4486-4492. 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), 102-129. doi: 10.1016/0022-247X(81)90095-0.  Google Scholar

[8]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, PA, 1990.  Google Scholar

[9]

G. Dai, Three solutions for a Neumann-type differential inclusion problem involving the $p(x)$-Laplacian, Nonlinear Anal., 70 (2009), 3755-3760. 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), 2297-2305. 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), 3064-3075. 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), 248-260. 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), 424-446. 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), 592-618.  Google Scholar

[15]

A. Kristály, Infinitely many solutions for a differential inclusion problem in $\mathbbR^n$, J. Differential Equations, 220 (2006), 511-530. 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), 367-379. 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), 37-52. 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), 108-120. 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), 1419-1425. 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), 2739-2743. 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), 757-774. 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, Application, Numerical Simulations, and Open Problems, Electronic Journal of Differential Equations Conference, 18, Southwest Texas State Univ., San Marcos, TX, (2010), 57-66.  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), 529-548. doi: 10.1088/0951-7715/23/3/005.  Google Scholar

[24]

B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.  Google Scholar

[25]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401–-410. doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar

show all references

References:
[1]

G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal., 54 (2003), 651-665. 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), 3031-3059. 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), 639-652. 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 - Fasc. I (2011), 105-113. 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), 1-10. 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), 4486-4492. 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), 102-129. doi: 10.1016/0022-247X(81)90095-0.  Google Scholar

[8]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, PA, 1990.  Google Scholar

[9]

G. Dai, Three solutions for a Neumann-type differential inclusion problem involving the $p(x)$-Laplacian, Nonlinear Anal., 70 (2009), 3755-3760. 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), 2297-2305. 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), 3064-3075. 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), 248-260. 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), 424-446. 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), 592-618.  Google Scholar

[15]

A. Kristály, Infinitely many solutions for a differential inclusion problem in $\mathbbR^n$, J. Differential Equations, 220 (2006), 511-530. 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), 367-379. 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), 37-52. 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), 108-120. 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), 1419-1425. 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), 2739-2743. 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), 757-774. 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, Application, Numerical Simulations, and Open Problems, Electronic Journal of Differential Equations Conference, 18, Southwest Texas State Univ., San Marcos, TX, (2010), 57-66.  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), 529-548. doi: 10.1088/0951-7715/23/3/005.  Google Scholar

[24]

B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.  Google Scholar

[25]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401–-410. doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar

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