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 & 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,, \emph{Ann. Univ. Ferrara Sez. VII Sci. Mat.}, 52 (2006), 19.  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,, \emph{Taiwanese Journal of Mathematics}, 5 (2011), 2291.   Google Scholar

[4]

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

[5]

M.-M. Boureanu and V. Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent,, \emph{Nonlinear Anal. TMA}, 75 (2012), 4471.  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,, \emph{Electron. J. Diff. Equ.}, 2013 (2013), 1.   Google Scholar

[7]

M.-M Boureanu and D.-N. Udrea, Existence and multiplicity result for elliptic problems with $p(\cdot)$-Growth conditions,, \emph{Nonlinear Anal.: Real World Applications}, 14 (2013), 1829.  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,, \emph{SIAM Journal of Applied Mathematics}, 66 (2006), 1383.  doi: 10.1137/050624522.  Google Scholar

[9]

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

[10]

X. Fan, Anisotropic variable exponent Sobolev spaces and $p(\cdot)$-Laplacian equations,, \emph{Complex Variables and Elliptic Equations}, 55 (2010), 1.  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,, \emph{Nonlinear Analysis TMA}, 67 (2007), 3064.  doi: 10.1016/j.na.2006.09.060.  Google Scholar

[12]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, \emph{J. Math. Anal. Appl.}, 263 (2001), 424.  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,, \emph{Discrete and Continuous Dynamical Systems-Series B}, 8 (2007), 61.  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,, \emph{Electronic Journal of Differential Equations}, 2009 (2009), 1.   Google Scholar

[16]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592.   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 (2010).  doi: 10.1017/CBO9780511760631.  Google Scholar

[18]

A. J. Kurdila and M. Zabarankin, Convex Functional Analysis,, Birkh\, (2005).   Google Scholar

[19]

V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces,, \emph{Nonlinear Anal. TMA}, 71 (2009), 3305.  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,, \emph{Electronic Journal of Differential Equations}, 2004 (2004), 1.   Google Scholar

[21]

Y. Liu, R. Davidson and P. Taylor, Investigation of the touch sensitivity of ER fluid based tactile display,, \emph{Proceedings of SPIE, 5764 (2005), 92.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity,, \emph{Math. USSR Izv.}, 29 (1987), 33.   Google Scholar

show all references

References:
[1]

S. N. Antontsev and J. F. Rodrigues, On stationary thermorheological viscous flows,, \emph{Ann. Univ. Ferrara Sez. VII Sci. Mat.}, 52 (2006), 19.  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,, \emph{Taiwanese Journal of Mathematics}, 5 (2011), 2291.   Google Scholar

[4]

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

[5]

M.-M. Boureanu and V. Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent,, \emph{Nonlinear Anal. TMA}, 75 (2012), 4471.  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,, \emph{Electron. J. Diff. Equ.}, 2013 (2013), 1.   Google Scholar

[7]

M.-M Boureanu and D.-N. Udrea, Existence and multiplicity result for elliptic problems with $p(\cdot)$-Growth conditions,, \emph{Nonlinear Anal.: Real World Applications}, 14 (2013), 1829.  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,, \emph{SIAM Journal of Applied Mathematics}, 66 (2006), 1383.  doi: 10.1137/050624522.  Google Scholar

[9]

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

[10]

X. Fan, Anisotropic variable exponent Sobolev spaces and $p(\cdot)$-Laplacian equations,, \emph{Complex Variables and Elliptic Equations}, 55 (2010), 1.  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,, \emph{Nonlinear Analysis TMA}, 67 (2007), 3064.  doi: 10.1016/j.na.2006.09.060.  Google Scholar

[12]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, \emph{J. Math. Anal. Appl.}, 263 (2001), 424.  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,, \emph{Discrete and Continuous Dynamical Systems-Series B}, 8 (2007), 61.  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,, \emph{Electronic Journal of Differential Equations}, 2009 (2009), 1.   Google Scholar

[16]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592.   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 (2010).  doi: 10.1017/CBO9780511760631.  Google Scholar

[18]

A. J. Kurdila and M. Zabarankin, Convex Functional Analysis,, Birkh\, (2005).   Google Scholar

[19]

V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces,, \emph{Nonlinear Anal. TMA}, 71 (2009), 3305.  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,, \emph{Electronic Journal of Differential Equations}, 2004 (2004), 1.   Google Scholar

[21]

Y. Liu, R. Davidson and P. Taylor, Investigation of the touch sensitivity of ER fluid based tactile display,, \emph{Proceedings of SPIE, 5764 (2005), 92.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity,, \emph{Math. USSR Izv.}, 29 (1987), 33.   Google Scholar

[1]

Inbo Sim, Yun-Ho Kim. Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents. Conference Publications, 2013, 2013 (special) : 695-707. doi: 10.3934/proc.2013.2013.695

[2]

Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066

[3]

Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201

[4]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080

[5]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439

[6]

F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355

[7]

SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026

[8]

Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129

[9]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683

[10]

M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián. Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. Communications on Pure & Applied Analysis, 2004, 3 (4) : 653-662. doi: 10.3934/cpaa.2004.3.653

[11]

Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179

[12]

M.J. Lopez-Herrero. The existence of weak solutions for a general class of mixed boundary value problems. Conference Publications, 2011, 2011 (Special) : 1015-1024. doi: 10.3934/proc.2011.2011.1015

[13]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[14]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

[15]

Mustapha Cheggag, Angelo Favini, Rabah Labbas, Stéphane Maingot, Ahmed Medeghri. Complete abstract differential equations of elliptic type with general Robin boundary conditions, in UMD spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 523-538. doi: 10.3934/dcdss.2011.4.523

[16]

Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273

[17]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729

[18]

Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037

[19]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113

[20]

Takahiro Hashimoto. Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients. Conference Publications, 2009, 2009 (Special) : 349-358. doi: 10.3934/proc.2009.2009.349

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]