-
Previous Article
Multiple solutions to a Neumann problem with equi-diffusive reaction term
- DCDS-S Home
- This Issue
-
Next Article
Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator
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), 651-665.
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), 3031-3059.
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), 639-652.
doi: 10.1002/mana.200810232. |
[4] |
G. Bonanno and A. Chinnì, Multiple solutions for elliptic problems involving the $p(x)$-Laplacian, Le Matematiche, LXVI - Fasc. I (2011), 105-113. |
[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. |
[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. |
[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. |
[8] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, PA, 1990. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math., 41 (1991), 592-618. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. |
[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. |
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. |
[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. |
[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. |
[4] |
G. Bonanno and A. Chinnì, Multiple solutions for elliptic problems involving the $p(x)$-Laplacian, Le Matematiche, LXVI - Fasc. I (2011), 105-113. |
[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. |
[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. |
[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. |
[8] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, PA, 1990. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math., 41 (1991), 592-618. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. |
[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. |
[1] |
Cristian Bereanu, Petru Jebelean. Multiple critical points for a class of periodic lower semicontinuous functionals. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 47-66. doi: 10.3934/dcds.2013.33.47 |
[2] |
Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615 |
[3] |
Keith Promislow, Hang Zhang. Critical points of functionalized Lagrangians. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1231-1246. doi: 10.3934/dcds.2013.33.1231 |
[4] |
Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357 |
[5] |
Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure and Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567 |
[6] |
Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217 |
[7] |
M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215 |
[8] |
Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231 |
[9] |
Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 |
[10] |
Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391 |
[11] |
P. Candito, S. A. Marano, D. Motreanu. Critical points for a class of nondifferentiable functions and applications. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 175-194. doi: 10.3934/dcds.2005.13.175 |
[12] |
Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327 |
[13] |
Stefano Almi, Massimo Fornasier, Richard Huber. Data-driven evolutions of critical points. Foundations of Data Science, 2020, 2 (3) : 207-255. doi: 10.3934/fods.2020011 |
[14] |
Marc Briane. Isotropic realizability of electric fields around critical points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 353-372. doi: 10.3934/dcdsb.2014.19.353 |
[15] |
Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 4215-4228. doi: 10.3934/era.2021080 |
[16] |
M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure and Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233 |
[17] |
Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907 |
[18] |
Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698 |
[19] |
Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645 |
[20] |
Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]