Liouville theorems for elliptic problems in variable exponent spaces

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March 2017, 16(2): 513-532. doi: 10.3934/cpaa.2017026

Liouville theorems for elliptic problems in variable exponent spaces

SYLWIA DUDEK ^1, and IWONA SKRZYPCZAK ^2,

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

Received May 2016 Revised November 2016 Published January 2017

We investigate nonexistence of nonnegative solutions to a partial differential inequality involving the p(x){Laplacian of the form

$- {\Delta _{p(x)}}u \geqslant \Phi (x,u(x),\nabla u(x))$

in ${\mathbb{R}^n}$, as well as in outer domain $\Omega \subseteq {\mathbb{R}^n}$, where Φ(x; u; ∇u) is a locally integrable Carathéodory’s function. We assume that Φ(x; u; ∇u) ≥ 0 or compatible with p and u. Growth conditions on u and p lead to Liouville-type results for u.

Keywords: Caccioppoli inequality, Liouville theorem, nonexistence of solutions, p(x)-Laplacian, p-Laplacian, variable exponent Lebesgue space.

Mathematics Subject Classification: 26D10, 35J60, 35J91.

Citation: 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

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. Hajłasz, 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. Kałamajska, 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 L^p(x)(Ω) and W^{m, 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. Hajłasz, 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. Kałamajska, 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 L^p(x)(Ω) and W^{m, 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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