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May  2015, 14(3): 881-896. doi: 10.3934/cpaa.2015.14.881

No--flux boundary value problems with anisotropic variable exponents

Maria-Magdalena Boureanu 1, and  Cristian Udrea 2,

1. 

Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585 Craiova
2. 

Department of Applied Mathematics, University of Craiova, A.I. Cuza Street 13, 200585 Craiova, Romania

Received  June 2014 Revised  November 2014 Published  March 2015

We are concerned with elliptic problems involving generalized anisotropic operators with variable exponents and a nonlinearity $f$. For such problems with no-flux boundary conditions we establish the existence, the uniqueness, or the multiplicity of weak solutions, under various hypotheses.
Keywords: Anisotropic variable exponents, uniqueness, Sobolev-type spaces, multiplicity., elliptic problems, no-flux boundary conditions, weak solutions, existence.
Mathematics Subject Classification: Primary: 35J25, 35D30, 35J20; Secondary: 46E3.
Citation: Maria-Magdalena Boureanu, Cristian Udrea. No--flux boundary value problems with anisotropic variable exponents. Communications on Pure & Applied Analysis, 2015, 14 (3) : 881-896. doi: 10.3934/cpaa.2015.14.881
References:
[1]

S. N. Antontsev and J. F. Rodrigues, On stationary thermorheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9.  Google Scholar

[2]

M.-M. Boureanu, A new class of general operators involved in anisotropic systems with variable exponents,, submitted., ().   Google Scholar

[3]

M.-M. Boureanu, Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese Journal of Mathematics, 5 (2011), 2291-2310.  Google Scholar

[4]

M.-M. Boureanu, A. Matei and M. Sofonea, Nonlinear problems with $p(\cdot)$-growth conditions and applications to antiplane contact models, Advanced Nonlinear Studies, 14 (2014), 295-313.  Google Scholar

[5]

M.-M. Boureanu and V. Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. TMA, 75 (2012), 4471-4482. doi: 10.1016/j.na.2011.09.033.  Google Scholar

[6]

M.-M Boureanu, C. Udrea and D.-N.Udrea, Anisotropic problems with variable exponents and constant Dirichlet condition, Electron. J. Diff. Equ., 2013 (2013), 1-13. Google Scholar

[7]

M.-M Boureanu and D.-N. Udrea, Existence and multiplicity result for elliptic problems with $p(\cdot)$-Growth conditions, Nonlinear Anal.: Real World Applications, 14 (2013), 1829-1844. doi: 10.1016/j.nonrwa.2012.12.001.  Google Scholar

[8]

Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal of Applied Mathematics, 66 (2006), 1383-1406. doi: 10.1137/050624522.  Google Scholar

[9]

D. G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4536-6.  Google Scholar

[10]

X. Fan, Anisotropic variable exponent Sobolev spaces and $p(\cdot)$-Laplacian equations, Complex Variables and Elliptic Equations, 55 (2010), 1-20. doi: 10.1080/17476931003728412.  Google Scholar

[11]

X. Fan and S.-G Deng, Remarks on Ricceri's variational principle and applications to the $p(x)-$Laplacian equations, Nonlinear Analysis TMA, 67 (2007), 3064-3075. doi: 10.1016/j.na.2006.09.060.  Google Scholar

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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

[13]

S. Gaucel and M. Langlais, Some remarks on a singular reaction-diffusion system arising in predator-prey modeling, Discrete and Continuous Dynamical Systems-Series B, 8 (2007), 61-72. doi: 10.3934/dcdsb.2007.8.61.  Google Scholar

[14]

Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Cambridge University Press, 2003. doi: 10.1017/CBO9780511546655.  Google Scholar

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B. Kone, S. Ouaro and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electronic Journal of Differential Equations, 2009 (2009), 1-11.  Google Scholar

[16]

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

[17]

A. Kristály, V. Rădulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.  Google Scholar

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A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkhäuser Verlag, 2005.  Google Scholar

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V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal. TMA, 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211.  Google Scholar

[20]

V. K. Le and K. Schmitt, Sub-supersolution theorens for quasilinear elliptic problems: a variational approach, Electronic Journal of Differential Equations, 2004 (2004), 1-7.  Google Scholar

[21]

Y. Liu, R. Davidson and P. Taylor, Investigation of the touch sensitivity of ER fluid based tactile display, Proceedings of SPIE, Smart Structures and Materials: Smart Structures and Integrated Systems, 5764 (2005), 92-99. Google Scholar

[22]

M. Mihăilescu and G. Moroşanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Applicable Analysis, 89 (2010), 257-271. doi: 10.1080/00036810802713826.  Google Scholar

[23]

J. Ovadia and Q. Nie, Stem cell niche structure as an inherent cause of undulating epithelial morphologies, Biophysical Journal, 104 (2013), 237-246. Google Scholar

[24]

P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. Series IX, 3 (2010), 543-584.  Google Scholar

[25]

N. Rodriguez, On the global well-posedness theory for a class of PDE models for criminal activity, Physica D, 260 (2013), 191-200. doi: 10.1016/j.physd.2012.08.003.  Google Scholar

[26]

M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002. Google Scholar

[27]

A. J. Simmonds, Electro-rheological valves in a hydraulic circuit, IEE Proceedings-D, 138 (1991), 400-404. Google Scholar

[28]

R. Stanway, J. L. Sproston and A. K. El-Wahed, Applications of electrorheological fluids in vibration control: a survey, Smart Mater. Struct., 5 (1996), 464-482. Google Scholar

[29]

L. Zhao, P. Zhao and X. Xie, Existence and multiplicity of solutions for divergence type elliptic equations, Electronic Journal of Differential Equations, 2011 (2011), 1-9. Google Scholar

[30]

V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33-66. Google Scholar

show all references

References:
[1]

S. N. Antontsev and J. F. Rodrigues, On stationary thermorheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9.  Google Scholar

[2]

M.-M. Boureanu, A new class of general operators involved in anisotropic systems with variable exponents,, submitted., ().   Google Scholar

[3]

M.-M. Boureanu, Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese Journal of Mathematics, 5 (2011), 2291-2310.  Google Scholar

[4]

M.-M. Boureanu, A. Matei and M. Sofonea, Nonlinear problems with $p(\cdot)$-growth conditions and applications to antiplane contact models, Advanced Nonlinear Studies, 14 (2014), 295-313.  Google Scholar

[5]

M.-M. Boureanu and V. Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. TMA, 75 (2012), 4471-4482. doi: 10.1016/j.na.2011.09.033.  Google Scholar

[6]

M.-M Boureanu, C. Udrea and D.-N.Udrea, Anisotropic problems with variable exponents and constant Dirichlet condition, Electron. J. Diff. Equ., 2013 (2013), 1-13. Google Scholar

[7]

M.-M Boureanu and D.-N. Udrea, Existence and multiplicity result for elliptic problems with $p(\cdot)$-Growth conditions, Nonlinear Anal.: Real World Applications, 14 (2013), 1829-1844. doi: 10.1016/j.nonrwa.2012.12.001.  Google Scholar

[8]

Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal of Applied Mathematics, 66 (2006), 1383-1406. doi: 10.1137/050624522.  Google Scholar

[9]

D. G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4536-6.  Google Scholar

[10]

X. Fan, Anisotropic variable exponent Sobolev spaces and $p(\cdot)$-Laplacian equations, Complex Variables and Elliptic Equations, 55 (2010), 1-20. doi: 10.1080/17476931003728412.  Google Scholar

[11]

X. Fan and S.-G Deng, Remarks on Ricceri's variational principle and applications to the $p(x)-$Laplacian equations, Nonlinear Analysis TMA, 67 (2007), 3064-3075. doi: 10.1016/j.na.2006.09.060.  Google Scholar

[12]

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

[13]

S. Gaucel and M. Langlais, Some remarks on a singular reaction-diffusion system arising in predator-prey modeling, Discrete and Continuous Dynamical Systems-Series B, 8 (2007), 61-72. doi: 10.3934/dcdsb.2007.8.61.  Google Scholar

[14]

Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Cambridge University Press, 2003. doi: 10.1017/CBO9780511546655.  Google Scholar

[15]

B. Kone, S. Ouaro and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electronic Journal of Differential Equations, 2009 (2009), 1-11.  Google Scholar

[16]

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

[17]

A. Kristály, V. Rădulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.  Google Scholar

[18]

A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkhäuser Verlag, 2005.  Google Scholar

[19]

V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal. TMA, 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211.  Google Scholar

[20]

V. K. Le and K. Schmitt, Sub-supersolution theorens for quasilinear elliptic problems: a variational approach, Electronic Journal of Differential Equations, 2004 (2004), 1-7.  Google Scholar

[21]

Y. Liu, R. Davidson and P. Taylor, Investigation of the touch sensitivity of ER fluid based tactile display, Proceedings of SPIE, Smart Structures and Materials: Smart Structures and Integrated Systems, 5764 (2005), 92-99. Google Scholar

[22]

M. Mihăilescu and G. Moroşanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Applicable Analysis, 89 (2010), 257-271. doi: 10.1080/00036810802713826.  Google Scholar

[23]

J. Ovadia and Q. Nie, Stem cell niche structure as an inherent cause of undulating epithelial morphologies, Biophysical Journal, 104 (2013), 237-246. Google Scholar

[24]

P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. Series IX, 3 (2010), 543-584.  Google Scholar

[25]

N. Rodriguez, On the global well-posedness theory for a class of PDE models for criminal activity, Physica D, 260 (2013), 191-200. doi: 10.1016/j.physd.2012.08.003.  Google Scholar

[26]

M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002. Google Scholar

[27]

A. J. Simmonds, Electro-rheological valves in a hydraulic circuit, IEE Proceedings-D, 138 (1991), 400-404. Google Scholar

[28]

R. Stanway, J. L. Sproston and A. K. El-Wahed, Applications of electrorheological fluids in vibration control: a survey, Smart Mater. Struct., 5 (1996), 464-482. Google Scholar

[29]

L. Zhao, P. Zhao and X. Xie, Existence and multiplicity of solutions for divergence type elliptic equations, Electronic Journal of Differential Equations, 2011 (2011), 1-9. Google Scholar

[30]

V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33-66. Google Scholar

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