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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 |
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. |
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. |
[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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