American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021029
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Positive solutions for the $p(x)-$Laplacian : Application of the Nehari method

 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 Early access March 2021

In this paper, we study the existence of positive solutions of the following equation
 $$$(P_{\lambda}) \left\{ \begin{array}{rclll} - \Delta_{p(x)} u+V(x)\vert u\vert^{p(x)-2}u & = & \lambda k(x) \vert u\vert^{\alpha(x)-2}u\\ &+& h(x) \vert u\vert^{\beta(x)-2}u&\mbox{ in }&\Omega\\ u& = &0 &\mbox{ on }& \partial \Omega. \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$$$
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
 $W_{0}^{1,p(x)}(\Omega)$
.
Citation: Said Taarabti. Positive solutions for the $p(x)-$Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [14] M. Dreher, The Kirchhoff equation for the $p$-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217-238.   Google Scholar [15] M. Dreher, The wave equation for the $p$-Laplacian, Hokkaido Math. J., 36 (2007), 21-52.  doi: 10.14492/hokmj/1285766660.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [25] O. Kováčik and J. Rǎkosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] W. Orlicz, $\ddot{U}$ber konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211.   Google Scholar [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.  Google Scholar [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.  Google Scholar [33] M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory, Lecture Note in Mathematics, 1748, Springer-Verlag, Berlin (2000). doi: 10.1007/BFb0104029.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [42] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv, 9 (1987), 33-66.   Google Scholar [43] V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.   Google Scholar [44] V. Zhikov, On passing to the limit in nonlinear variational problem, Math. Sb., 183 (1992), 47-84.  doi: 10.1070/SM1993v076n02ABEH003421.  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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [14] M. Dreher, The Kirchhoff equation for the $p$-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217-238.   Google Scholar [15] M. Dreher, The wave equation for the $p$-Laplacian, Hokkaido Math. J., 36 (2007), 21-52.  doi: 10.14492/hokmj/1285766660.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [25] O. Kováčik and J. Rǎkosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] W. Orlicz, $\ddot{U}$ber konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211.   Google Scholar [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.  Google Scholar [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.  Google Scholar [33] M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory, Lecture Note in Mathematics, 1748, Springer-Verlag, Berlin (2000). doi: 10.1007/BFb0104029.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [42] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv, 9 (1987), 33-66.   Google Scholar [43] V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.   Google Scholar [44] V. Zhikov, On passing to the limit in nonlinear variational problem, Math. Sb., 183 (1992), 47-84.  doi: 10.1070/SM1993v076n02ABEH003421.  Google Scholar
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