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Harnack inequality for degenerate elliptic equations and sum operators
The Liouville theorems for elliptic equations with nonstandard growth
1. | Institute of Mathematics of the Polish Academy of Sciences, 00-956 Warsaw, Poland |
2. | Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland |
References:
[1] |
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. |
[2] |
T. Adamowicz, Phragmén-Lindelöf theorems for equations with nonstandard growth, Nonlinear Anal., 97 (2014), 169-184.
doi: 10.1016/j.na.2013.11.018. |
[3] |
L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869.
doi: 10.1016/j.na.2008.12.028. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[7] |
F. Cammaroto and L. Vilasi, On a perturbed $p(x)$-Laplacian problem in bounded and unbounded domains, J. Math. Anal. Appl., 402 (2013), 71-83.
doi: 10.1016/j.jmaa.2013.01.013. |
[8] |
G. Caristi and E. Mitidieri, Some Liouville theorems for quasilinear elliptic inequalities, Doklady Math., 79 (2009), 118-124.
doi: 10.1134/S1064562409010360. |
[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] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Berlin-Heidelberg 2011.
doi: 10.1007/978-3-642-18363-8. |
[11] |
L. Diening and M. Růžička, Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 413-450.
doi: 10.1007/s00021-004-0124-8. |
[12] |
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. |
[13] |
X.-L. Fan and X. Fan, A Knobloch-type result for $p(t)$-Laplacian systems, J. Math. Anal. Appl., 282 (2003), 453-464.
doi: 10.1016/S0022-247X(02)00376-1. |
[14] |
X.-L. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417.
doi: 10.1016/j.jde.2007.01.008. |
[15] |
R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.
doi: 10.1016/j.na.2008.12.018. |
[16] |
Y. Fu, Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain, Topol. Methods Nonlinear Anal., 30 (2007), 235-249. |
[17] |
P. Hästö, On the existence of minimizers of the variable exponent Dirichlet energy integral, Commun. Pure Appl. Anal., 5 (2006), 413-420.
doi: 10.3934/cpaa.2006.5.415. |
[18] |
P. Harjulehto, P. Hästö, Ût V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
[19] |
P. Harjulehto, P. Hästö, M. Koskenoja, T. Lukkari and N. Marola, An obstacle problem and superharmonic functions with nonstandard growth, Nonlinear Anal., 67 (2007), 3424-3440.
doi: 10.1016/j.na.2006.10.026. |
[20] |
J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0131-8. |
[21] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006. |
[22] |
I. Holopainen and P. Pankka, $p$-Laplace operator, quasiregular mappings and Picard-type theorems, Quasiconformal mappings and their applications, 117-150, Narosa, New Delhi, 2007. |
[23] |
P. Lindqvist, On the growth of the solutions of the differential equation div$(|\nabla u|^{p -2} \nabla u ) = 0$ in $n$-dimensional space, J. Differential Equations, 58 (1985), 307-317.
doi: 10.1016/0022-0396(85)90002-6. |
[24] |
W. Liu and P. Zhao, Existence of positive solutions for $p(x)$-Laplacian equations in unbounded domains, Nonlinear Anal., 69 (2008), 3358-3371.
doi: 10.1016/j.na.2007.09.027. |
[25] |
O. Martio, Quasiminimizing properties of solutions to Riccati type equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 823-832. |
[26] |
E. Mitidieri and S. I. Pokhozaev, Some generalizations of the Bernstein Theorem, Differential Equations, 38 (2002), 373-378.
doi: 10.1023/A:1016066010721. |
[27] |
N.-C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations, 35 (2010), 1958-1981.
doi: 10.1080/03605300903585344. |
[28] |
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. |
[29] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748 Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[30] |
J. Serrin, The Liouville theorem for homogeneous elliptic differential inequalities. Problems in mathematical analysis, No. 61. J. Math. Sci. (N. Y.), 179 (2011), 174-183.
doi: 10.1007/s10958-011-0588-z. |
[31] |
L. F. Wang, Liouville theorem for the variable exponent Laplacian (Chinese), J. East China Norm. Univ. Natur. Sci. Ed., 1 (2009), 84-93. |
[32] |
V. Zhikov, On some variational problems (Russian), J. Math. Phys., 5 (1997), 105-116 (1998). |
[33] |
V. Zhikov, Density of smooth functions in Sobolev-Orlicz spaces, J. Math. Sci. (N. Y.), 132 (2006), 285-294.
doi: 10.1007/s10958-005-0497-0. |
show all references
References:
[1] |
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. |
[2] |
T. Adamowicz, Phragmén-Lindelöf theorems for equations with nonstandard growth, Nonlinear Anal., 97 (2014), 169-184.
doi: 10.1016/j.na.2013.11.018. |
[3] |
L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869.
doi: 10.1016/j.na.2008.12.028. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[7] |
F. Cammaroto and L. Vilasi, On a perturbed $p(x)$-Laplacian problem in bounded and unbounded domains, J. Math. Anal. Appl., 402 (2013), 71-83.
doi: 10.1016/j.jmaa.2013.01.013. |
[8] |
G. Caristi and E. Mitidieri, Some Liouville theorems for quasilinear elliptic inequalities, Doklady Math., 79 (2009), 118-124.
doi: 10.1134/S1064562409010360. |
[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] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Berlin-Heidelberg 2011.
doi: 10.1007/978-3-642-18363-8. |
[11] |
L. Diening and M. Růžička, Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 413-450.
doi: 10.1007/s00021-004-0124-8. |
[12] |
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. |
[13] |
X.-L. Fan and X. Fan, A Knobloch-type result for $p(t)$-Laplacian systems, J. Math. Anal. Appl., 282 (2003), 453-464.
doi: 10.1016/S0022-247X(02)00376-1. |
[14] |
X.-L. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417.
doi: 10.1016/j.jde.2007.01.008. |
[15] |
R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.
doi: 10.1016/j.na.2008.12.018. |
[16] |
Y. Fu, Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain, Topol. Methods Nonlinear Anal., 30 (2007), 235-249. |
[17] |
P. Hästö, On the existence of minimizers of the variable exponent Dirichlet energy integral, Commun. Pure Appl. Anal., 5 (2006), 413-420.
doi: 10.3934/cpaa.2006.5.415. |
[18] |
P. Harjulehto, P. Hästö, Ût V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
[19] |
P. Harjulehto, P. Hästö, M. Koskenoja, T. Lukkari and N. Marola, An obstacle problem and superharmonic functions with nonstandard growth, Nonlinear Anal., 67 (2007), 3424-3440.
doi: 10.1016/j.na.2006.10.026. |
[20] |
J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0131-8. |
[21] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006. |
[22] |
I. Holopainen and P. Pankka, $p$-Laplace operator, quasiregular mappings and Picard-type theorems, Quasiconformal mappings and their applications, 117-150, Narosa, New Delhi, 2007. |
[23] |
P. Lindqvist, On the growth of the solutions of the differential equation div$(|\nabla u|^{p -2} \nabla u ) = 0$ in $n$-dimensional space, J. Differential Equations, 58 (1985), 307-317.
doi: 10.1016/0022-0396(85)90002-6. |
[24] |
W. Liu and P. Zhao, Existence of positive solutions for $p(x)$-Laplacian equations in unbounded domains, Nonlinear Anal., 69 (2008), 3358-3371.
doi: 10.1016/j.na.2007.09.027. |
[25] |
O. Martio, Quasiminimizing properties of solutions to Riccati type equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 823-832. |
[26] |
E. Mitidieri and S. I. Pokhozaev, Some generalizations of the Bernstein Theorem, Differential Equations, 38 (2002), 373-378.
doi: 10.1023/A:1016066010721. |
[27] |
N.-C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations, 35 (2010), 1958-1981.
doi: 10.1080/03605300903585344. |
[28] |
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. |
[29] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748 Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[30] |
J. Serrin, The Liouville theorem for homogeneous elliptic differential inequalities. Problems in mathematical analysis, No. 61. J. Math. Sci. (N. Y.), 179 (2011), 174-183.
doi: 10.1007/s10958-011-0588-z. |
[31] |
L. F. Wang, Liouville theorem for the variable exponent Laplacian (Chinese), J. East China Norm. Univ. Natur. Sci. Ed., 1 (2009), 84-93. |
[32] |
V. Zhikov, On some variational problems (Russian), J. Math. Phys., 5 (1997), 105-116 (1998). |
[33] |
V. Zhikov, Density of smooth functions in Sobolev-Orlicz spaces, J. Math. Sci. (N. Y.), 132 (2006), 285-294.
doi: 10.1007/s10958-005-0497-0. |
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