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Liouville theorems for elliptic problems in variable exponent spaces
1. | Institute of Mathematics, Krakow University of Technology, ul. Warszawska 24, 31-155 Krakow, Poland |
2. | Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland |
$- {\Delta _{p(x)}}u \geqslant \Phi (x,u(x),\nabla u(x))$ |
References:
[1] |
E. Acerbi, I. Fonseca and G. Mingione, Existence and regularity for mixtures of micromagnetic materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2225-2243.
doi: 10.1098/rspa.2006.1655. |
[2] |
E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.
doi: 10.1007/s002050100117. |
[3] |
E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259.
doi: 10.1007/s00205-002-0208-7. |
[4] |
T. Adamowicz and P. Górka, The Liouville theorems for elliptic equations with nonstandard growth, Commun. Pure Appl. Anal., 14 (2015), 2377-2392.
doi: 10.3934/cpaa.2015.14.2377. |
[5] |
A. Baalal and A. Qabil, Liouville-type result for quasilinear elliptic problems with variable exponent, Int. J. Pure Appl. Math., 104 (2015), 57-68. |
[6] |
S. Barnás, Existence results for hemivariational inequality involving p(x)-Laplacian, Opuscula Math., 32 (2012), 439-454.
doi: 10.7494/OpMath.2012.32.3.439. |
[7] |
A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J.-L. Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431-436.
doi: 10.1016/j.crma.2007.01.011. |
[8] |
B. Bojarski and P. Hałlasz, Pointwise inequalities for Sobolev functions and some applications, Studia Math., 106 (1993), 77-92. |
[9] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[10] |
D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013.
doi: 10.1007/978-3-0348-0548-3. |
[11] |
L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869.
doi: 10.1016/j.na.2008.12.028. |
[12] |
L. D'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020.
doi: 10.1016/j.aim.2009.12.017. |
[13] |
L. D'Ambrosio and E. Mitidieri, A priori estimates and reduction principles for quasilinear elliptic problems and applications, Adv. Differential Equations, 17 (2012), 935-1000. |
[14] |
L. D'Ambrosio and E. Mitidieri, Liouville theorems for elliptic systems and applications, J. Math. Anal. Appl., 413 (2014), 121-138.
doi: 10.1016/j.jmaa.2013.11.052. |
[15] |
R. N. Dhara and A. Ka lamajska, On equivalent conditions for the validity of Poincaré inequality on weighted Sobolev space with applications to the solvability of degenerated PDEs involving p-Laplacian, J. Math. Anal. Appl., 432 (2015), 463-483.
doi: 10.1016/j.jmaa.2015.06.068. |
[16] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Springer, Heidelberg, 2011. Lecture Notes in Mathematics, 2017. |
[17] |
T.-L. Dinu, Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent, Nonlinear Anal., 65 (2006), 1414-1424.
doi: 10.1016/j.na.2005.10.022. |
[18] |
O. Došlý and R. Mařík, Nonexistence of positive solutions of PDE's with p-Laplacian, Acta Math. Hungar., 90 (2001), 89-107.
doi: 10.1023/A:1006739909182. |
[19] |
P. Drábek, A. Ka lamajska and I. Skrzypczak, Caccioppoli-type estimates and Hardy-type inequalities derived from degenerated p-harmonic problems, preprint, 2016. |
[20] |
X. Fan, On the positive solutions of p(x)-laplace equation, J. Gansu Educ. College, 15 (2001), 401-407. |
[21] |
X. Fan, Positive solutions to p(x)-Laplacian-Dirichlet problems with sign-changing nonlinearities, Glasg. Math. J., 52 (2010), 505-516.
doi: 10.1017/S0017089510000388. |
[22] |
X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm, p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
[23] |
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. |
[24] |
R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.
doi: 10.1016/j.na.2008.12.018. |
[25] |
E. Galakhov, O. Salieva and L. Uvarova, Nonexistence results for some nonlinear elliptic and parabolic inequalities with functional parameters, Electron. J. Qual. Theory Differ. Equ., 85 (2015), 1-11. |
[26] |
N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.
doi: 10.1007/s00208-010-0510-x. |
[27] |
P. Gwiazda, F. Z. Klawe and A. Świerczewska-Gwiazda, Thermo-visco-elasticity for NortonHoff-type models, Nonlinear Anal. Real World Appl., 26 (2015), 199-228.
doi: 10.1016/j.nonrwa.2015.05.009. |
[28] |
P. Gwiazda, P. Minakowski and A. Wróblewska-Kamińska, Elliptic problems in generalized Orlicz-Musielak spaces, Cent. Eur. J. Math., 10 (2012), 2019-2032.
doi: 10.2478/s11533-012-0126-3. |
[29] |
P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Math. Methods Appl. Sci., 33 (2010), 125-137.
doi: 10.1002/mma.1155. |
[30] |
P. Harjulehto, P. Hästö and V. Latvala, Harnack's inequality for p(·)-harmonic functions with unbounded exponent p, J. Math. Anal. Appl., 352 (2009), 345-359.
doi: 10.1016/j.jmaa.2008.05.090. |
[31] |
P. Harjulehto, P. Hästö, Ú. V. Lê and M. Nuortio, Overview of differential equations with nonstandard growth, Nonlinear Anal., 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
[32] |
P. Harjulehto, J. Kinnunen and T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl., 2007. Art. ID 48348. |
[33] |
P. S. Iliaş, Existence and multiplicity of solutions of a p(x)-Laplacian equation in a bounded domain, Rev. Roumaine Math. Pures Appl., 52 (2007), 639-653. |
[34] |
Y. Jiang and Y. Fu, On the eigenvalue of p(x)-laplace equation, arXiv: 1105.4225v1, 2011. |
[35] |
A. Kałamajska, K. Pietruska-Pałuba and I. Skrzypczak, Nonexistence results for differential inequalities involving A-Laplacian, Adv. Differential Equations, 17 (2012), 307-336. |
[36] |
G. Karisti, E. Mitidieri and S. I. Pokhozhaev, Liouville theorems for quasilinear elliptic inequalities, Dokl. Akad. Nauk, 424 (2009), 741-747.
doi: 10.1134/S1064562409010360. |
[37] |
J. Liu, Positive solutions of the p(x)-Laplace equation with singular nonlinearity, Nonlinear Anal., 72 (2010), 4428-4437.
doi: 10.1016/j.na.2010.02.018. |
[38] |
M. Mihăilescu and D. Stancu-Dumitru, On an eigenvalue problem involving the p(x)-Laplace operator plus a non-local term, Differ. Equ. Appl., 1 (2009), 367-378.
doi: 10.7153/dea-01-20. |
[39] |
E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in ${{\mathbb{R}}^{n}}$, Tr. Mat. Inst. Steklova, 227 (1999), 192-222. |
[40] |
E. Mitidieri and S. I. Pokhozhaev, Some generalizations of Bernstein's theorem, Differ. Uravn., 38 (2002), 373-378.
doi: 10.1023/A:1016066010721. |
[41] |
P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.
doi: 10.1016/j.jde.2014.05.023. |
[42] |
K. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Res. Commun., 1996,401-407. |
[43] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. Lecture Notes in Mathematics, 1748.
doi: 10.1007/BFb0104029. |
[44] |
J. Serrin, The Liouville theorem for homogeneous elliptic differential inequalities. Problems in mathematical analysis, 61, J. Math. Sci. (N. Y.), 179 (2011), 174-183.
doi: 10.1007/s10958-011-0588-z. |
[45] |
I. Skrzypczak, Hardy-type inequalities derived from p-harmonic problems, Nonlinear Anal., 93 (2013), 30-50.
doi: 10.1016/j.na.2013.07.006. |
[46] |
I. Skrzypczak, Hardy-Poincaré type inequalities derived from p-harmonic problems, In Calculus of variations and PDEs, Banach Center Publ. 101, pages 225-238. Polish Acad. Sci. Inst. Math. , Warsaw, 2014.
doi: 10.4064/bc101-0-17. |
[47] |
L. F. Wang, Liouville theorem for the variable exponent Laplacian, J. East China Norm. Univ. Natur. Sci. Ed., 1 (2009), 84-93. |
[48] |
V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system, Differ. Uravn., 33 (1997), 107-114. |
[49] |
V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116. |
show all references
References:
[1] |
E. Acerbi, I. Fonseca and G. Mingione, Existence and regularity for mixtures of micromagnetic materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2225-2243.
doi: 10.1098/rspa.2006.1655. |
[2] |
E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.
doi: 10.1007/s002050100117. |
[3] |
E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259.
doi: 10.1007/s00205-002-0208-7. |
[4] |
T. Adamowicz and P. Górka, The Liouville theorems for elliptic equations with nonstandard growth, Commun. Pure Appl. Anal., 14 (2015), 2377-2392.
doi: 10.3934/cpaa.2015.14.2377. |
[5] |
A. Baalal and A. Qabil, Liouville-type result for quasilinear elliptic problems with variable exponent, Int. J. Pure Appl. Math., 104 (2015), 57-68. |
[6] |
S. Barnás, Existence results for hemivariational inequality involving p(x)-Laplacian, Opuscula Math., 32 (2012), 439-454.
doi: 10.7494/OpMath.2012.32.3.439. |
[7] |
A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J.-L. Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431-436.
doi: 10.1016/j.crma.2007.01.011. |
[8] |
B. Bojarski and P. Hałlasz, Pointwise inequalities for Sobolev functions and some applications, Studia Math., 106 (1993), 77-92. |
[9] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[10] |
D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013.
doi: 10.1007/978-3-0348-0548-3. |
[11] |
L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869.
doi: 10.1016/j.na.2008.12.028. |
[12] |
L. D'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020.
doi: 10.1016/j.aim.2009.12.017. |
[13] |
L. D'Ambrosio and E. Mitidieri, A priori estimates and reduction principles for quasilinear elliptic problems and applications, Adv. Differential Equations, 17 (2012), 935-1000. |
[14] |
L. D'Ambrosio and E. Mitidieri, Liouville theorems for elliptic systems and applications, J. Math. Anal. Appl., 413 (2014), 121-138.
doi: 10.1016/j.jmaa.2013.11.052. |
[15] |
R. N. Dhara and A. Ka lamajska, On equivalent conditions for the validity of Poincaré inequality on weighted Sobolev space with applications to the solvability of degenerated PDEs involving p-Laplacian, J. Math. Anal. Appl., 432 (2015), 463-483.
doi: 10.1016/j.jmaa.2015.06.068. |
[16] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Springer, Heidelberg, 2011. Lecture Notes in Mathematics, 2017. |
[17] |
T.-L. Dinu, Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent, Nonlinear Anal., 65 (2006), 1414-1424.
doi: 10.1016/j.na.2005.10.022. |
[18] |
O. Došlý and R. Mařík, Nonexistence of positive solutions of PDE's with p-Laplacian, Acta Math. Hungar., 90 (2001), 89-107.
doi: 10.1023/A:1006739909182. |
[19] |
P. Drábek, A. Ka lamajska and I. Skrzypczak, Caccioppoli-type estimates and Hardy-type inequalities derived from degenerated p-harmonic problems, preprint, 2016. |
[20] |
X. Fan, On the positive solutions of p(x)-laplace equation, J. Gansu Educ. College, 15 (2001), 401-407. |
[21] |
X. Fan, Positive solutions to p(x)-Laplacian-Dirichlet problems with sign-changing nonlinearities, Glasg. Math. J., 52 (2010), 505-516.
doi: 10.1017/S0017089510000388. |
[22] |
X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm, p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
[23] |
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. |
[24] |
R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.
doi: 10.1016/j.na.2008.12.018. |
[25] |
E. Galakhov, O. Salieva and L. Uvarova, Nonexistence results for some nonlinear elliptic and parabolic inequalities with functional parameters, Electron. J. Qual. Theory Differ. Equ., 85 (2015), 1-11. |
[26] |
N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.
doi: 10.1007/s00208-010-0510-x. |
[27] |
P. Gwiazda, F. Z. Klawe and A. Świerczewska-Gwiazda, Thermo-visco-elasticity for NortonHoff-type models, Nonlinear Anal. Real World Appl., 26 (2015), 199-228.
doi: 10.1016/j.nonrwa.2015.05.009. |
[28] |
P. Gwiazda, P. Minakowski and A. Wróblewska-Kamińska, Elliptic problems in generalized Orlicz-Musielak spaces, Cent. Eur. J. Math., 10 (2012), 2019-2032.
doi: 10.2478/s11533-012-0126-3. |
[29] |
P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Math. Methods Appl. Sci., 33 (2010), 125-137.
doi: 10.1002/mma.1155. |
[30] |
P. Harjulehto, P. Hästö and V. Latvala, Harnack's inequality for p(·)-harmonic functions with unbounded exponent p, J. Math. Anal. Appl., 352 (2009), 345-359.
doi: 10.1016/j.jmaa.2008.05.090. |
[31] |
P. Harjulehto, P. Hästö, Ú. V. Lê and M. Nuortio, Overview of differential equations with nonstandard growth, Nonlinear Anal., 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
[32] |
P. Harjulehto, J. Kinnunen and T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl., 2007. Art. ID 48348. |
[33] |
P. S. Iliaş, Existence and multiplicity of solutions of a p(x)-Laplacian equation in a bounded domain, Rev. Roumaine Math. Pures Appl., 52 (2007), 639-653. |
[34] |
Y. Jiang and Y. Fu, On the eigenvalue of p(x)-laplace equation, arXiv: 1105.4225v1, 2011. |
[35] |
A. Kałamajska, K. Pietruska-Pałuba and I. Skrzypczak, Nonexistence results for differential inequalities involving A-Laplacian, Adv. Differential Equations, 17 (2012), 307-336. |
[36] |
G. Karisti, E. Mitidieri and S. I. Pokhozhaev, Liouville theorems for quasilinear elliptic inequalities, Dokl. Akad. Nauk, 424 (2009), 741-747.
doi: 10.1134/S1064562409010360. |
[37] |
J. Liu, Positive solutions of the p(x)-Laplace equation with singular nonlinearity, Nonlinear Anal., 72 (2010), 4428-4437.
doi: 10.1016/j.na.2010.02.018. |
[38] |
M. Mihăilescu and D. Stancu-Dumitru, On an eigenvalue problem involving the p(x)-Laplace operator plus a non-local term, Differ. Equ. Appl., 1 (2009), 367-378.
doi: 10.7153/dea-01-20. |
[39] |
E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in ${{\mathbb{R}}^{n}}$, Tr. Mat. Inst. Steklova, 227 (1999), 192-222. |
[40] |
E. Mitidieri and S. I. Pokhozhaev, Some generalizations of Bernstein's theorem, Differ. Uravn., 38 (2002), 373-378.
doi: 10.1023/A:1016066010721. |
[41] |
P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.
doi: 10.1016/j.jde.2014.05.023. |
[42] |
K. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Res. Commun., 1996,401-407. |
[43] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. Lecture Notes in Mathematics, 1748.
doi: 10.1007/BFb0104029. |
[44] |
J. Serrin, The Liouville theorem for homogeneous elliptic differential inequalities. Problems in mathematical analysis, 61, J. Math. Sci. (N. Y.), 179 (2011), 174-183.
doi: 10.1007/s10958-011-0588-z. |
[45] |
I. Skrzypczak, Hardy-type inequalities derived from p-harmonic problems, Nonlinear Anal., 93 (2013), 30-50.
doi: 10.1016/j.na.2013.07.006. |
[46] |
I. Skrzypczak, Hardy-Poincaré type inequalities derived from p-harmonic problems, In Calculus of variations and PDEs, Banach Center Publ. 101, pages 225-238. Polish Acad. Sci. Inst. Math. , Warsaw, 2014.
doi: 10.4064/bc101-0-17. |
[47] |
L. F. Wang, Liouville theorem for the variable exponent Laplacian, J. East China Norm. Univ. Natur. Sci. Ed., 1 (2009), 84-93. |
[48] |
V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system, Differ. Uravn., 33 (1997), 107-114. |
[49] |
V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116. |
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