April  2018, 11(2): 179-191. doi: 10.3934/dcdss.2018011

One-dimensional nonlinear boundary value problems with variable exponent

Department of Engineering, University of Messina, 98166 Messina -Italy

* Corresponding author: G. Bonanno

Received  February 2017 Revised  May 2017 Published  January 2018

In this paper, a class of nonlinear differential boundary value problems with variable exponent is investigated. The existence of at least one non-zero solution is established, without assuming on the nonlinear term any condition either at zero or at infinity. The approach is developed within the framework of the Orlicz-Sobolev spaces with variable exponent and it is based on a local minimum theorem for differentiable functions.

Citation: Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011
References:
[1]

R. AboulaichD. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.  doi: 10.1016/j.camwa.2008.01.017.  Google Scholar

[2]

G. Barletta and A. Chinní, Existence of solutions for a Neumann problem involving the $p(x)-$Laplacian, Electron. J. Differential Equations, 2013 (2013), 1-12.   Google Scholar

[3]

G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75 (2012), 2992-3007.  doi: 10.1016/j.na.2011.12.003.  Google Scholar

[4]

G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205-220.  doi: 10.1515/anona-2012-0003.  Google Scholar

[5]

G. Bonanno and A. Chinní, Discontinuous elliptic problems involving the p(x)-Laplacian, Math. Nachr., 284 (2011), 639-652.  doi: 10.1002/mana.200810232.  Google Scholar

[6]

G. Bonanno and A. Chinní, Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl., 418 (2014), 812-827.  doi: 10.1016/j.jmaa.2014.04.016.  Google Scholar

[7]

D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces Applied and Numerical Harmonic Analysis, Springer Basel, Heidelberg, 2013. doi: 10.1007/978-3-0348-0548-3.  Google Scholar

[8]

G. D'Aguí and A. Sciammetta, Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions, Nonlinear Anal., 75 (2012), 5612-5619.  doi: 10.1016/j.na.2012.05.009.  Google Scholar

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 2017. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[10]

X.-L. FanJ. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(Ω)$, J. Math. Anal. Appl., 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.  Google Scholar

[11]

X.-L. Fan and D. Zhao, On the spaces $L^{p(x)}(Ω)$ and $W^{m,p(x)}(Ω)$, J. Math. Anal., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[12]

X. -L. Fan, Some results on variable exponent analysis, More Progresses in Analysis, Proceedings of the 5th International ISAAC Congress, World Scientific, New Jersey, 2009, 93–99. doi: 10.1142/9789812835635_0008.  Google Scholar

[13]

X.-L. Fan and Q.-H. Zhang, Existence of solutions for $p(x)-$Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[14]

X.-L. 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

[15]

O. Kováčik and J. Rákosník, On the spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math., 41 (1991), 592-618.   Google Scholar

[16]

J. Musielak, Orlicz Spaces and Modular Spaces Lecture Notes in Mathematics 1034, Springer, Berlin, 1983. doi: 10.1007/BFb0072210.  Google Scholar

[17]

M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory Springer-Verlag, Berlin, 2000. Google Scholar

show all references

References:
[1]

R. AboulaichD. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.  doi: 10.1016/j.camwa.2008.01.017.  Google Scholar

[2]

G. Barletta and A. Chinní, Existence of solutions for a Neumann problem involving the $p(x)-$Laplacian, Electron. J. Differential Equations, 2013 (2013), 1-12.   Google Scholar

[3]

G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75 (2012), 2992-3007.  doi: 10.1016/j.na.2011.12.003.  Google Scholar

[4]

G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205-220.  doi: 10.1515/anona-2012-0003.  Google Scholar

[5]

G. Bonanno and A. Chinní, Discontinuous elliptic problems involving the p(x)-Laplacian, Math. Nachr., 284 (2011), 639-652.  doi: 10.1002/mana.200810232.  Google Scholar

[6]

G. Bonanno and A. Chinní, Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl., 418 (2014), 812-827.  doi: 10.1016/j.jmaa.2014.04.016.  Google Scholar

[7]

D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces Applied and Numerical Harmonic Analysis, Springer Basel, Heidelberg, 2013. doi: 10.1007/978-3-0348-0548-3.  Google Scholar

[8]

G. D'Aguí and A. Sciammetta, Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions, Nonlinear Anal., 75 (2012), 5612-5619.  doi: 10.1016/j.na.2012.05.009.  Google Scholar

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 2017. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[10]

X.-L. FanJ. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(Ω)$, J. Math. Anal. Appl., 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.  Google Scholar

[11]

X.-L. Fan and D. Zhao, On the spaces $L^{p(x)}(Ω)$ and $W^{m,p(x)}(Ω)$, J. Math. Anal., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[12]

X. -L. Fan, Some results on variable exponent analysis, More Progresses in Analysis, Proceedings of the 5th International ISAAC Congress, World Scientific, New Jersey, 2009, 93–99. doi: 10.1142/9789812835635_0008.  Google Scholar

[13]

X.-L. Fan and Q.-H. Zhang, Existence of solutions for $p(x)-$Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[14]

X.-L. 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

[15]

O. Kováčik and J. Rákosník, On the spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math., 41 (1991), 592-618.   Google Scholar

[16]

J. Musielak, Orlicz Spaces and Modular Spaces Lecture Notes in Mathematics 1034, Springer, Berlin, 1983. doi: 10.1007/BFb0072210.  Google Scholar

[17]

M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory Springer-Verlag, Berlin, 2000. Google Scholar

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