# American Institute of Mathematical Sciences

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

## No--flux boundary value problems with anisotropic variable exponents

 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.
Citation: Maria-Magdalena Boureanu, Cristian Udrea. No--flux boundary value problems with anisotropic variable exponents. Communications on Pure and 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. [2] M.-M. Boureanu, A new class of general operators involved in anisotropic systems with variable exponents, submitted. [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. [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. [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. [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. [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. [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. [9] D. G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4536-6. [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. [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. [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. [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. [14] Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Cambridge University Press, 2003. doi: 10.1017/CBO9780511546655. [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. [16] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618. [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. [18] A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkhäuser Verlag, 2005. [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. [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. [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. [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. [23] J. Ovadia and Q. Nie, Stem cell niche structure as an inherent cause of undulating epithelial morphologies, Biophysical Journal, 104 (2013), 237-246. [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. [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. [26] M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002. [27] A. J. Simmonds, Electro-rheological valves in a hydraulic circuit, IEE Proceedings-D, 138 (1991), 400-404. [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. [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. [30] V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33-66.

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##### 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. [2] M.-M. Boureanu, A new class of general operators involved in anisotropic systems with variable exponents, submitted. [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. [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. [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. [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. [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. [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. [9] D. G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4536-6. [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. [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. [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. [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. [14] Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Cambridge University Press, 2003. doi: 10.1017/CBO9780511546655. [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. [16] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618. [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. [18] A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkhäuser Verlag, 2005. [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. [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. [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. [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. [23] J. Ovadia and Q. Nie, Stem cell niche structure as an inherent cause of undulating epithelial morphologies, Biophysical Journal, 104 (2013), 237-246. [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. [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. [26] M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002. [27] A. J. Simmonds, Electro-rheological valves in a hydraulic circuit, IEE Proceedings-D, 138 (1991), 400-404. [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. [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. [30] V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33-66.
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