International Journal of Nonlinear Analysis and Applications
$L^\infty$-regularity result for an obstacle problem with degenerate coercivity in Musielak-Sobolev spaces
Document Type : Research Paper
Authors
1 Laboratory of Mathematical Analysis and Applications (LAMA), Department of Mathematics, Faculty of Sciences Dhar el Mahraz, Sidi Mohamed Ben Abdellah University, PB 1796 Fez, Morocco
2 Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, P.O.Box 20, 24000 El Jadida, Morocco
Abstract
Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N},$ $N\geq 2$. In this paper we give an existence result of bounded solution, in Musielak spaces, for unilateral problems associated to the nonlinear elliptic equation $$
-\mathop{\rm div}a(x,u,{\nabla}u)+g(x,u,\nabla u)=f \quad\text{in }{\Omega},$$ where the nonlinearity $g$ does not satisfy the well known sign condition and $f$ is an integrable source.
Keywords
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[2] L. Aharouch, A. Benkirane and M. Rhoudaf, Strongly nonlinear elliptic variational unilateral problems in Orlicz
spaces, Abstr. Appl. Anal. 2006 (2006), 1–20.
[3] L. Aharouch, A. Benkirane, M. Rhoudaf, Existence results for some unilateral problems without sign condition
with obstacle free in Orlicz spaces, Nonlinear Anal. 68 (2008), no. 8. 2362–2380.
[4] L. Aharouch and M. Rhoudaf, Strongly nonlinear elliptic unilateral problems in Orlicz space and L
1
-data, J.
Inequal. Pure Appl. Math. 6 (2005), no. 2, Art. 54, 1–20.
[5] Y. Ahmida, I. Chlebicka, P. Gwiazda and A. Youssfi, Gossez’s approximation theorems in Musielak-Orlicz-Sobolev
spaces, J. Funct. Anal. 275 (2018), no. 9, 2538–2571.
[6] M. Ait Khellou and A. Benkirane:Correction to: Elliptic inequalities with L
1 data in MusielakOrlicz spaces,
Monat. Math. 187 (2018), no. 1, 181–187.
[7] M. Ait Khellou, A. Benkirane and S.M. Douiri: Existence of solutions for elliptic equations having natural growth
terms in Musielak-Orlicz spaces, J. Math. Comput. Sci. 4 (2014), no. 4, 665–688.
[8] M. Ait Khellou, A. Benkirane and S.M. Douiri, Some properties of Musielak spaces with only the log-Hlder
continuity condition and application, Ann. Funct. Anal. 11 (2020), no. 4, 1062–1080.
[9] Y. Akdim, A. Benkirane and M. El Moumni, Solutions of nonlin- ear elliptic problems with lower order terms,
Ann. Funct. Anal. 6 (2015), no. 1, 34–53.
[10] A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations
with degenerate coercivity, Ann. Mat. Pura Appl., IV. Ser. 182 (2003), no.1, 53–79.
[11] A. Alvino, V. Ferone and G. Trombetti, A priori estimates for a class of non uniformly elliptic equations, Atti
Semin. Mat. Fis. Univ. Modena 46 (1998), 381–391.
[12] A. Benkirane and M.S. El Vally(Ould Mohameden Val): Some approximation properties in Musielak-OrliczSobolev spaces, Thai. J. Math. 10 (2012), no. 2, 371–381.
[13] A. Benkirane, M.S. El Vally, An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces,
Bull. Belg. Math.Soc. Simon Stevin 20 (2013), no. 1, 57–75.
[14] A. Benkirane and M.S. El Vally, Variational inequalities in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math. Soc.
Simon Stevin, 21 (2014), no. 5, 787–811.
[15] A. Benkirane and A. Youssfi, existence of bounded solutions for a class of strongly nonlinear elliptic equations in
Orlicz-Sobolev spaces, Aust. J. Math. Anal. 5 (2008), no. 1, Art. 7, 1–26.
[16] A. Benkirane and A. Youssfi, existence of bounded sollutions for a class of strongly nonlinear elliptic equations in
Orlicz-Sobolev spaces, Aust. J. Math. Anal. 5 (2008), no. 1, Art. 7, 1–26.
[17] A. Benkirane, A. Youssfi and M. El Moumni, Bounded solutions of unilateral problems for strongly nonlinear
elliptic equations in Orlicz spaces, Elect. J. of Qual. Th. of Diff. Eq. 2013 (2013), no. 21, 1–25.
[18] C. Bennett, R. Sharpley: Interpolation of operators, Academic press, Boston, 1988.
[19] L. Boccardo, A. Dall’Aglio, L. Orsina: Existence and regularity results for some elliptic equations with degenerate
coercivity. Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 51–81.
[20] L. Boccardo, F. Murat, J. P. Puel; L∞ estimate for some nonlinear elliptic partial differential equations and
application to an existence result, SIAM J. Math. Anal. 2 (1992), 326–333.
[21] L. Boccardo, S. Segura de le´on and C. Trombetti, Bounded and unbounded solution for a class of quasi-linear
elliptic problems with aquadratic gradient term, J. Math. Pures Appl. 80 (2001), 919–940.
[22] L. Boccardo, F. Murat and J.P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems,
Ann. Mat. Pura Appl. 152 (1988), 183–196.
[23] L. Boccardo and G. Rita Cirmi: Unilateral problems with degenerate coercivity, Le Matematiche Vol. LIV Sup-plemento, (1999) 6173.
[24] M. Bourahma, A. Benkirane and J. Bennouna, Existence of renormalized solutions for some nonlinear elliptic
equations in Orlicz spaces, J. Rend. Circ. Mat. Palermo, II. Ser 2 69 (2020), no. 1, 231–252.
[25] M. Bourahma, A. Benkirane and J. Bennouna,An existence result of entropy solutions to elliptic problems in
generalized Orlicz-Sobolev spaces, J. Rend. Circ. Mat. Palermo, II. Ser 2 70 (2021), no. 1, 481–504.
[26] M. Bourahma, J. Bennouna and M. El Moumni: Existence of a weak bounded solutions for a nonlinear degenerate
elliptic equations in Musielak spaces, Moroccan J. Pure and Appl. Anal. 6 (2020), no. 1, 16–33.
[27] I. Chlebicka, P. Gwiazda and A. Zatorska Goldstein: Well-posedness of parabolic equations in the non-reflexive
and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions, 265 (2018), no. 11, 5716–5766.
[28] R. Elarabi, M. Rhoudaf, H. Sabiki: Entropy solution for a nonlinear elliptic problem with lower order term in
Musielak-Orlicz spaces, Ric. Mat. 67 (2018), no. 2, 549–579.
[29] A. Elmahi and D. Meskine: Nonlinear elliptic problems having natural growth and L
1 data in Orlicz spaces, Ann.
Mat. Pura Appl. 184 (2005), no. 2, 161–184.
[30] A. Elmahi and D. Meskine: Elliptic inequalities with lower order terms and L
1 data in Orlicz spaces. J. Math.
Anal. Appl. 328 (2007), 1417-1434.
[31] V. Ferone, M.R. Posteraro and J.M. Rakotosone, L∞ estimates for nonlinear elliptic problem with p-growth in
the gradient, J. Ineq. Appl. 3 (1999), 109–125.
[32] J.P. Gossez: A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, Proc. Sympos. Pure Math. 45, Amer.
Math. Soc., (1986), 455–462.
[33] J. P. Gossez: Surjectivity results for pseudo-monotone mappings in complementary systems, J. Math. Anal. Appl.
53 (1976), 484–494.
[34] J.P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163–205.
[35] J.P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Anal. 11 (1987), 379-49.
[36] P. Gwiazda, I. Skrzypczak and A. Zatorska.Goldstein, Existence of renormalized solutions to elliptic equation in
Musielak-Orlicz space, J. Differential Equ. 264 (2018), 341-377.
[37] P. Harjulehto and P. H¨ast¨o, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, vol.
2236, Springer, Cham, 2019.
[38] M. Krasnosel’skii and Ya. Rutikii, Convex functions and Orlicz spaces, Groningen, Nordhooff, 1969.
[39] J. Musielak: Orlicz spaces and modular spaces, Lecture Note in Mahtematics, 1034, Springer, Berlin, 1983.
[40] G. Talenti, Nonlinear elliptic equations, Rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl.
IV. Ser. 120 (1979), 159–184.
[41] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 697–718.
[42] G. Talenti, Linear elliptic P.D.E’s: Level sets, rearrangements and a priori estimates of solutions, Boll. Un. Mat.
Ital. 4 (1985), no. 6, 917–949.
[43] C. Trombetti, Non-uniformly elliptic equations with natural growth in the gradient, Potential Anal. 18 (2003),
391–404.
[44] A. Youssfi, Existence of bounded solution for nonlinear degenerate elliptic equations in Orlicz spaces, Electron. J.
Differ. Equ. 2007 (2007), no. 54, 1–13.
[2] L. Aharouch, A. Benkirane and M. Rhoudaf, Strongly nonlinear elliptic variational unilateral problems in Orlicz
spaces, Abstr. Appl. Anal. 2006 (2006), 1–20.
[3] L. Aharouch, A. Benkirane, M. Rhoudaf, Existence results for some unilateral problems without sign condition
with obstacle free in Orlicz spaces, Nonlinear Anal. 68 (2008), no. 8. 2362–2380.
[4] L. Aharouch and M. Rhoudaf, Strongly nonlinear elliptic unilateral problems in Orlicz space and L
1
-data, J.
Inequal. Pure Appl. Math. 6 (2005), no. 2, Art. 54, 1–20.
[5] Y. Ahmida, I. Chlebicka, P. Gwiazda and A. Youssfi, Gossez’s approximation theorems in Musielak-Orlicz-Sobolev
spaces, J. Funct. Anal. 275 (2018), no. 9, 2538–2571.
[6] M. Ait Khellou and A. Benkirane:Correction to: Elliptic inequalities with L
1 data in MusielakOrlicz spaces,
Monat. Math. 187 (2018), no. 1, 181–187.
[7] M. Ait Khellou, A. Benkirane and S.M. Douiri: Existence of solutions for elliptic equations having natural growth
terms in Musielak-Orlicz spaces, J. Math. Comput. Sci. 4 (2014), no. 4, 665–688.
[8] M. Ait Khellou, A. Benkirane and S.M. Douiri, Some properties of Musielak spaces with only the log-Hlder
continuity condition and application, Ann. Funct. Anal. 11 (2020), no. 4, 1062–1080.
[9] Y. Akdim, A. Benkirane and M. El Moumni, Solutions of nonlin- ear elliptic problems with lower order terms,
Ann. Funct. Anal. 6 (2015), no. 1, 34–53.
[10] A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations
with degenerate coercivity, Ann. Mat. Pura Appl., IV. Ser. 182 (2003), no.1, 53–79.
[11] A. Alvino, V. Ferone and G. Trombetti, A priori estimates for a class of non uniformly elliptic equations, Atti
Semin. Mat. Fis. Univ. Modena 46 (1998), 381–391.
[12] A. Benkirane and M.S. El Vally(Ould Mohameden Val): Some approximation properties in Musielak-OrliczSobolev spaces, Thai. J. Math. 10 (2012), no. 2, 371–381.
[13] A. Benkirane, M.S. El Vally, An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces,
Bull. Belg. Math.Soc. Simon Stevin 20 (2013), no. 1, 57–75.
[14] A. Benkirane and M.S. El Vally, Variational inequalities in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math. Soc.
Simon Stevin, 21 (2014), no. 5, 787–811.
[15] A. Benkirane and A. Youssfi, existence of bounded solutions for a class of strongly nonlinear elliptic equations in
Orlicz-Sobolev spaces, Aust. J. Math. Anal. 5 (2008), no. 1, Art. 7, 1–26.
[16] A. Benkirane and A. Youssfi, existence of bounded sollutions for a class of strongly nonlinear elliptic equations in
Orlicz-Sobolev spaces, Aust. J. Math. Anal. 5 (2008), no. 1, Art. 7, 1–26.
[17] A. Benkirane, A. Youssfi and M. El Moumni, Bounded solutions of unilateral problems for strongly nonlinear
elliptic equations in Orlicz spaces, Elect. J. of Qual. Th. of Diff. Eq. 2013 (2013), no. 21, 1–25.
[18] C. Bennett, R. Sharpley: Interpolation of operators, Academic press, Boston, 1988.
[19] L. Boccardo, A. Dall’Aglio, L. Orsina: Existence and regularity results for some elliptic equations with degenerate
coercivity. Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 51–81.
[20] L. Boccardo, F. Murat, J. P. Puel; L∞ estimate for some nonlinear elliptic partial differential equations and
application to an existence result, SIAM J. Math. Anal. 2 (1992), 326–333.
[21] L. Boccardo, S. Segura de le´on and C. Trombetti, Bounded and unbounded solution for a class of quasi-linear
elliptic problems with aquadratic gradient term, J. Math. Pures Appl. 80 (2001), 919–940.
[22] L. Boccardo, F. Murat and J.P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems,
Ann. Mat. Pura Appl. 152 (1988), 183–196.
[23] L. Boccardo and G. Rita Cirmi: Unilateral problems with degenerate coercivity, Le Matematiche Vol. LIV Sup-plemento, (1999) 6173.
[24] M. Bourahma, A. Benkirane and J. Bennouna, Existence of renormalized solutions for some nonlinear elliptic
equations in Orlicz spaces, J. Rend. Circ. Mat. Palermo, II. Ser 2 69 (2020), no. 1, 231–252.
[25] M. Bourahma, A. Benkirane and J. Bennouna,An existence result of entropy solutions to elliptic problems in
generalized Orlicz-Sobolev spaces, J. Rend. Circ. Mat. Palermo, II. Ser 2 70 (2021), no. 1, 481–504.
[26] M. Bourahma, J. Bennouna and M. El Moumni: Existence of a weak bounded solutions for a nonlinear degenerate
elliptic equations in Musielak spaces, Moroccan J. Pure and Appl. Anal. 6 (2020), no. 1, 16–33.
[27] I. Chlebicka, P. Gwiazda and A. Zatorska Goldstein: Well-posedness of parabolic equations in the non-reflexive
and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions, 265 (2018), no. 11, 5716–5766.
[28] R. Elarabi, M. Rhoudaf, H. Sabiki: Entropy solution for a nonlinear elliptic problem with lower order term in
Musielak-Orlicz spaces, Ric. Mat. 67 (2018), no. 2, 549–579.
[29] A. Elmahi and D. Meskine: Nonlinear elliptic problems having natural growth and L
1 data in Orlicz spaces, Ann.
Mat. Pura Appl. 184 (2005), no. 2, 161–184.
[30] A. Elmahi and D. Meskine: Elliptic inequalities with lower order terms and L
1 data in Orlicz spaces. J. Math.
Anal. Appl. 328 (2007), 1417-1434.
[31] V. Ferone, M.R. Posteraro and J.M. Rakotosone, L∞ estimates for nonlinear elliptic problem with p-growth in
the gradient, J. Ineq. Appl. 3 (1999), 109–125.
[32] J.P. Gossez: A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, Proc. Sympos. Pure Math. 45, Amer.
Math. Soc., (1986), 455–462.
[33] J. P. Gossez: Surjectivity results for pseudo-monotone mappings in complementary systems, J. Math. Anal. Appl.
53 (1976), 484–494.
[34] J.P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163–205.
[35] J.P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Anal. 11 (1987), 379-49.
[36] P. Gwiazda, I. Skrzypczak and A. Zatorska.Goldstein, Existence of renormalized solutions to elliptic equation in
Musielak-Orlicz space, J. Differential Equ. 264 (2018), 341-377.
[37] P. Harjulehto and P. H¨ast¨o, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, vol.
2236, Springer, Cham, 2019.
[38] M. Krasnosel’skii and Ya. Rutikii, Convex functions and Orlicz spaces, Groningen, Nordhooff, 1969.
[39] J. Musielak: Orlicz spaces and modular spaces, Lecture Note in Mahtematics, 1034, Springer, Berlin, 1983.
[40] G. Talenti, Nonlinear elliptic equations, Rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl.
IV. Ser. 120 (1979), 159–184.
[41] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 697–718.
[42] G. Talenti, Linear elliptic P.D.E’s: Level sets, rearrangements and a priori estimates of solutions, Boll. Un. Mat.
Ital. 4 (1985), no. 6, 917–949.
[43] C. Trombetti, Non-uniformly elliptic equations with natural growth in the gradient, Potential Anal. 18 (2003),
391–404.
[44] A. Youssfi, Existence of bounded solution for nonlinear degenerate elliptic equations in Orlicz spaces, Electron. J.
Differ. Equ. 2007 (2007), no. 54, 1–13.
)
Volume 13, Issue 2
July 2022Pages 1617-1641
- Receive Date: 11 August 2021
- Revise Date: 30 January 2022
- Accept Date: 08 May 2022