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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 |
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. 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. |
[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. |
show all references
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. 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. |
[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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