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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 |
$\begin{equation} (P_{\lambda}) \left\{ \begin{array}{rclll} - \Delta_{p(x)} u+V(x)\vert u\vert^{p(x)-2}u & = & \lambda k(x) \vert u\vert^{\alpha(x)-2}u\\ &+& h(x) \vert u\vert^{\beta(x)-2}u&\mbox{ in }&\Omega\\ u& = &0 &\mbox{ on }& \partial \Omega. \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$ |
$ (P_{\lambda}) $ |
$ (P_{\lambda}) $ |
$ W_{0}^{1,p(x)}(\Omega) $ |
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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