American Institute of Mathematical Sciences

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November  2015, 14(6): 2377-2392. doi: 10.3934/cpaa.2015.14.2377

The Liouville theorems for elliptic equations with nonstandard growth

Tomasz Adamowicz 1, and  Przemysław Górka 2,

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

Received  February 2015 Revised  June 2015 Published  September 2015

We study solutions and supersolutions of homogeneous and nonhomogeneous $A$-harmonic equations with nonstandard growth in $\mathbb{R}^n$. Various Liouville-type theorems and nonexistence results are proved. The discussion is illustrated by a number of examples.
Keywords: $p$-harmonic, $p$-Laplace, $A$-harmonic, $p(x)$-supersolution, Phragmén-Lindelöf theorem, Riccati equation, variable exponent., $p(x)$-harmonic, nonstandard growth, Liouville theorem.
Mathematics Subject Classification: Primary 35B53; Secondary: 35J92, 46E3.
Citation: Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377
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.  Google Scholar

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R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916. doi: 10.1016/j.na.2008.12.018.  Google Scholar

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Y. Fu, Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain, Topol. Methods Nonlinear Anal., 30 (2007), 235-249.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[25]

O. Martio, Quasiminimizing properties of solutions to Riccati type equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 823-832.  Google Scholar

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E. Mitidieri and S. I. Pokhozaev, Some generalizations of the Bernstein Theorem, Differential Equations, 38 (2002), 373-378. doi: 10.1023/A:1016066010721.  Google Scholar

[27]

N.-C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations, 35 (2010), 1958-1981. doi: 10.1080/03605300903585344.  Google Scholar

[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.  Google Scholar

[29]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748 Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[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.  Google Scholar

[31]

L. F. Wang, Liouville theorem for the variable exponent Laplacian (Chinese), J. East China Norm. Univ. Natur. Sci. Ed., 1 (2009), 84-93.  Google Scholar

[32]

V. Zhikov, On some variational problems (Russian), J. Math. Phys., 5 (1997), 105-116 (1998).  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869. doi: 10.1016/j.na.2008.12.028.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Caristi and E. Mitidieri, Some Liouville theorems for quasilinear elliptic inequalities, Doklady Math., 79 (2009), 118-124. doi: 10.1134/S1064562409010360.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. Fu, Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain, Topol. Methods Nonlinear Anal., 30 (2007), 235-249.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.  Google Scholar

[21]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

O. Martio, Quasiminimizing properties of solutions to Riccati type equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 823-832.  Google Scholar

[26]

E. Mitidieri and S. I. Pokhozaev, Some generalizations of the Bernstein Theorem, Differential Equations, 38 (2002), 373-378. doi: 10.1023/A:1016066010721.  Google Scholar

[27]

N.-C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations, 35 (2010), 1958-1981. doi: 10.1080/03605300903585344.  Google Scholar

[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.  Google Scholar

[29]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748 Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[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.  Google Scholar

[31]

L. F. Wang, Liouville theorem for the variable exponent Laplacian (Chinese), J. East China Norm. Univ. Natur. Sci. Ed., 1 (2009), 84-93.  Google Scholar

[32]

V. Zhikov, On some variational problems (Russian), J. Math. Phys., 5 (1997), 105-116 (1998).  Google Scholar

[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.  Google Scholar

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