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January  2022, 15(1): 229-243. doi: 10.3934/dcdss.2021029

Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method

Said Taarabti ,

Laboratory of Systems Engineering and Information Technologies (LISTI), National School of Applied Sciences of Agadir, Ibn Zohr University, Morocco

* Corresponding author: Said Taarabti (s.taarabti@uiz.ac.ma)

Received  August 2020 Revised  February 2021 Published  January 2022 Early access  March 2021

In this paper, we study the existence of positive solutions of the following equation
$\begin{equation} (P_{\lambda}) \left\{ \begin{array}{rclll} - \Delta_{p(x)} u+V(x)\vert u\vert^{p(x)-2}u & = & \lambda k(x) \vert u\vert^{\alpha(x)-2}u\\ &+& h(x) \vert u\vert^{\beta(x)-2}u&\mbox{ in }&\Omega\\ u& = &0 &\mbox{ on }& \partial \Omega. \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$
The study of the problem
$ (P_{\lambda}) $
 needs generalized Lebesgue and Sobolev spaces. In this work, under suitable assumptions, we prove that some variational methods still work. We use them to prove the existence of positive solutions to the problem
$ (P_{\lambda}) $
 in
$ W_{0}^{1,p(x)}(\Omega) $
.
Keywords: p(x)-Laplacian, Nehari manifold, Dirichlet boundary conditions.
Mathematics Subject Classification: Primary: 35B38, 35D05; Secondary: 35J60.
Citation: Said Taarabti. Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 229-243. doi: 10.3934/dcdss.2021029
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]

G. A. Afrouzi, S. Mahdavi and Z. Naghizadeh, The Nehari manifold for $p-$Laplacian equation with Dirichlet boundary condition, Nonlinear Anal. Model. Control, 12 (2007). doi: 10.15388/NA.2007.12.2.14705.

[3]

C. O. Alves and J. L. P. Barreiro, Existence and multiplicity of solutions for a $p(x)-$Laplacian equation with critical growth, J. Math. Anal. Appl., 403 (2013), 143-154.  doi: 10.1016/j.jmaa.2013.02.025.

[4]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[5]

S. Antontsev and S. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localisation properties of solutions, 60 (2005), 515–545. doi: 10.1016/j.na.2004.09.026.

[6]

C. O. Alves, M. A. S. Souto, Existence of solutions for a class of problems in $ \mathbb{R}^N $ involving the $p(x)$-Laplacian, Nonlinear Differential Equations Appl., 66 (2006), 17-32. doi: 10.1007/3-7643-7401-2_2.

[7]

B. Cekic and R. A. Mashiyev, Existence and localization results for $p(x)$-Laplacian via topological methods, Fixed Point Theory Appl., (2010), Art. ID 120646. doi: 10.1155/2010/120646.

[8]

J. Chabrowski and Y. Fu, Existence of solutions for $p(x)$-Laplacian problems on a bounded domain, J. Math. Anal. Appl., 306 (2005), 604-618.  doi: 10.1016/j.jmaa.2004.10.028.

[9]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[10]

F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 263-277.  doi: 10.1017/S000497270003570X.

[11]

J. P. P. Da Silva, On some multiple solutions for a $p(x)$-Laplacian equation with critical growth, J. Math. Anal. Appl., 436 (2016), 782-795.  doi: 10.1016/j.jmaa.2015.11.078.

[12]

L. Diening, P. Hasto and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, FSDONA04 Proceedings, Milovy, Czech Republic, (2004), 38–58.

[13]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p-$Laplacian: Application of the fibrering method, Proceedings of the Royal Society of Edinburgh, 127A (1997) 703–726. doi: 10.1017/S0308210500023787.

[14]

M. Dreher, The Kirchhoff equation for the $p$-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217-238. 

[15]

M. Dreher, The wave equation for the $p$-Laplacian, Hokkaido Math. J., 36 (2007), 21-52.  doi: 10.14492/hokmj/1285766660.

[16]

D. E. Edmunds and J. Rákosník, Density of smooth functions in $W^{k, p(x)}(\Omega)$, Proc. Roy. Soc. London Ser. A, 437 (1992), 229-236.  doi: 10.1098/rspa.1992.0059.

[17]

D. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Stud. Math., 143 (2000) 267–293. doi: 10.4064/sm-143-3-267-293.

[18]

X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl, 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.

[19]

L. Fan and D. Zhao, On the spaces $L^{p(x)}$ and $W^{m; p(x)}$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[20]

X. L. Fan, On the sub-supersolution method for $p(x)-$Laplacian equations, J. Math. Anal. Appl, 330 (2007) 665–682. doi: 10.1016/j.jmaa.2006.07.093.

[21]

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.

[22]

X. L. Fan, Q. H. Zhang and D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.  doi: 10.1016/j.jmaa.2003.11.020.

[23]

X. L. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for $p(x)-$Laplace equations (in Chinese), Chinese Ann. Math. Ser. A, 24 (2003), 495-500. 

[24]

R. Kajikia, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005) 352–370. doi: 10.1016/j.jfa.2005.04.005.

[25]

O. Kováčik and J. Rǎkosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618. 

[26]

H. Lane, On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, Am. J. Sci., 50 (1869), 57-74.  doi: 10.1016/B978-0-08-006653-0.50032-3.

[27]

A. Marcos and A. Abdou, A. Existence of solutions for a nonhomogeneous Dirichlet problem involving $p(x)$-Laplacian operator and indefinite weight, Bound Value Probl., 171 (2019). doi: 10.1186/s13661-019-1276-z.

[28]

M. Mihailescu and V. Radulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.  doi: 10.1090/S0002-9939-07-08815-6.

[29]

R. A. Mashiyev, S. Ogras, Z. Yucedag and M. Avci, The Nehari manifold approach for Dirichlet prioblem involving the $p(x)-$laplacien equation, J. Korean Math. Soc., 47 (2010), 845-860.  doi: 10.4134/JKMS.2010.47.4.845.

[30]

W. Orlicz, $\ddot{U}$ber konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211. 

[31]

S. H. Rasouli and K. Fallah, The Nehari manifold approach for a $p(x)-$Laplacien problem with nonlinear boundary conditions, Ukrainian Mathematical Journal, 69, 2017, 92–103. doi: 10.1007/s11253-017-1350-6.

[32]

V. Rǎdulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.  doi: 10.1016/j.na.2011.01.037.

[33]

M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory, Lecture Note in Mathematics, 1748, Springer-Verlag, Berlin (2000). doi: 10.1007/BFb0104029.

[34]

S. Saiedinezhad and M. B. Ghaemi, The fibering map approach to a quasilinear degenerate $p(x)-$Laplacian equation, Bull. Iranian Math. Soc., 41 (2015), 1477-1492. 

[35]

K. Saoudi, Existence and nonexistence of positive solutions for quasilinear elliptic problem, Abstract and Applied Analysis, (2012), Art. ID 275748. doi: 10.1155/2012/275748.

[36]

K. Saoudi, Existence and multiplicity of solutions for a quasilinear equation involving the $p(x)$-Laplace operator, Complex Variables and Elliptic Equations, 62 (2017), 318-332.  doi: 10.1080/17476933.2016.1219999.

[37]

S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators, Integral Transforms Spec. Funct., 16 (2005), 461-482.  doi: 10.1080/10652460412331320322.

[38]

A. Silva, Multiple solutions for the $p(x)$-Laplace operator with critical growth, Adv. Nonlinear Stud., 11 (2011), 63-75.  doi: 10.1515/ans-2011-0103.

[39]

N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.

[40]

Z. Y$\ddot{u}$cedaǧ, Solutions of nonlinear problems involving p(x)-Laplacian operator, Adv. Nonlinear Anal., 4 (2015), 285-293.  doi: 10.1515/anona-2015-0044.

[41]

X. Zhang and X. Liu, The local boundedness and Harnack inequality of p(x)-Laplace equation, J. Math. Anal. Appl., 332 (2007), 209-218.  doi: 10.1016/j.jmaa.2006.10.021.

[42]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv, 9 (1987), 33-66. 

[43]

V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710. 

[44]

V. Zhikov, On passing to the limit in nonlinear variational problem, Math. Sb., 183 (1992), 47-84.  doi: 10.1070/SM1993v076n02ABEH003421.

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]

G. A. Afrouzi, S. Mahdavi and Z. Naghizadeh, The Nehari manifold for $p-$Laplacian equation with Dirichlet boundary condition, Nonlinear Anal. Model. Control, 12 (2007). doi: 10.15388/NA.2007.12.2.14705.

[3]

C. O. Alves and J. L. P. Barreiro, Existence and multiplicity of solutions for a $p(x)-$Laplacian equation with critical growth, J. Math. Anal. Appl., 403 (2013), 143-154.  doi: 10.1016/j.jmaa.2013.02.025.

[4]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[5]

S. Antontsev and S. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localisation properties of solutions, 60 (2005), 515–545. doi: 10.1016/j.na.2004.09.026.

[6]

C. O. Alves, M. A. S. Souto, Existence of solutions for a class of problems in $ \mathbb{R}^N $ involving the $p(x)$-Laplacian, Nonlinear Differential Equations Appl., 66 (2006), 17-32. doi: 10.1007/3-7643-7401-2_2.

[7]

B. Cekic and R. A. Mashiyev, Existence and localization results for $p(x)$-Laplacian via topological methods, Fixed Point Theory Appl., (2010), Art. ID 120646. doi: 10.1155/2010/120646.

[8]

J. Chabrowski and Y. Fu, Existence of solutions for $p(x)$-Laplacian problems on a bounded domain, J. Math. Anal. Appl., 306 (2005), 604-618.  doi: 10.1016/j.jmaa.2004.10.028.

[9]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[10]

F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 263-277.  doi: 10.1017/S000497270003570X.

[11]

J. P. P. Da Silva, On some multiple solutions for a $p(x)$-Laplacian equation with critical growth, J. Math. Anal. Appl., 436 (2016), 782-795.  doi: 10.1016/j.jmaa.2015.11.078.

[12]

L. Diening, P. Hasto and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, FSDONA04 Proceedings, Milovy, Czech Republic, (2004), 38–58.

[13]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p-$Laplacian: Application of the fibrering method, Proceedings of the Royal Society of Edinburgh, 127A (1997) 703–726. doi: 10.1017/S0308210500023787.

[14]

M. Dreher, The Kirchhoff equation for the $p$-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217-238. 

[15]

M. Dreher, The wave equation for the $p$-Laplacian, Hokkaido Math. J., 36 (2007), 21-52.  doi: 10.14492/hokmj/1285766660.

[16]

D. E. Edmunds and J. Rákosník, Density of smooth functions in $W^{k, p(x)}(\Omega)$, Proc. Roy. Soc. London Ser. A, 437 (1992), 229-236.  doi: 10.1098/rspa.1992.0059.

[17]

D. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Stud. Math., 143 (2000) 267–293. doi: 10.4064/sm-143-3-267-293.

[18]

X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl, 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.

[19]

L. Fan and D. Zhao, On the spaces $L^{p(x)}$ and $W^{m; p(x)}$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[20]

X. L. Fan, On the sub-supersolution method for $p(x)-$Laplacian equations, J. Math. Anal. Appl, 330 (2007) 665–682. doi: 10.1016/j.jmaa.2006.07.093.

[21]

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.

[22]

X. L. Fan, Q. H. Zhang and D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.  doi: 10.1016/j.jmaa.2003.11.020.

[23]

X. L. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for $p(x)-$Laplace equations (in Chinese), Chinese Ann. Math. Ser. A, 24 (2003), 495-500. 

[24]

R. Kajikia, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005) 352–370. doi: 10.1016/j.jfa.2005.04.005.

[25]

O. Kováčik and J. Rǎkosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618. 

[26]

H. Lane, On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, Am. J. Sci., 50 (1869), 57-74.  doi: 10.1016/B978-0-08-006653-0.50032-3.

[27]

A. Marcos and A. Abdou, A. Existence of solutions for a nonhomogeneous Dirichlet problem involving $p(x)$-Laplacian operator and indefinite weight, Bound Value Probl., 171 (2019). doi: 10.1186/s13661-019-1276-z.

[28]

M. Mihailescu and V. Radulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.  doi: 10.1090/S0002-9939-07-08815-6.

[29]

R. A. Mashiyev, S. Ogras, Z. Yucedag and M. Avci, The Nehari manifold approach for Dirichlet prioblem involving the $p(x)-$laplacien equation, J. Korean Math. Soc., 47 (2010), 845-860.  doi: 10.4134/JKMS.2010.47.4.845.

[30]

W. Orlicz, $\ddot{U}$ber konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211. 

[31]

S. H. Rasouli and K. Fallah, The Nehari manifold approach for a $p(x)-$Laplacien problem with nonlinear boundary conditions, Ukrainian Mathematical Journal, 69, 2017, 92–103. doi: 10.1007/s11253-017-1350-6.

[32]

V. Rǎdulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.  doi: 10.1016/j.na.2011.01.037.

[33]

M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory, Lecture Note in Mathematics, 1748, Springer-Verlag, Berlin (2000). doi: 10.1007/BFb0104029.

[34]

S. Saiedinezhad and M. B. Ghaemi, The fibering map approach to a quasilinear degenerate $p(x)-$Laplacian equation, Bull. Iranian Math. Soc., 41 (2015), 1477-1492. 

[35]

K. Saoudi, Existence and nonexistence of positive solutions for quasilinear elliptic problem, Abstract and Applied Analysis, (2012), Art. ID 275748. doi: 10.1155/2012/275748.

[36]

K. Saoudi, Existence and multiplicity of solutions for a quasilinear equation involving the $p(x)$-Laplace operator, Complex Variables and Elliptic Equations, 62 (2017), 318-332.  doi: 10.1080/17476933.2016.1219999.

[37]

S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators, Integral Transforms Spec. Funct., 16 (2005), 461-482.  doi: 10.1080/10652460412331320322.

[38]

A. Silva, Multiple solutions for the $p(x)$-Laplace operator with critical growth, Adv. Nonlinear Stud., 11 (2011), 63-75.  doi: 10.1515/ans-2011-0103.

[39]

N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.

[40]

Z. Y$\ddot{u}$cedaǧ, Solutions of nonlinear problems involving p(x)-Laplacian operator, Adv. Nonlinear Anal., 4 (2015), 285-293.  doi: 10.1515/anona-2015-0044.

[41]

X. Zhang and X. Liu, The local boundedness and Harnack inequality of p(x)-Laplace equation, J. Math. Anal. Appl., 332 (2007), 209-218.  doi: 10.1016/j.jmaa.2006.10.021.

[42]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv, 9 (1987), 33-66. 

[43]

V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710. 

[44]

V. Zhikov, On passing to the limit in nonlinear variational problem, Math. Sb., 183 (1992), 47-84.  doi: 10.1070/SM1993v076n02ABEH003421.

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