ENTROPY SOLUTIONS TO NONLINEAR ELLIPTIC ANISOTROPIC PROBLEM WITH VARIABLE EXPONENT

Mohamed Badr Benboubker; Hassane Hjiaj; Stanislas Ouaro

doi:10.11948/2014012

2014 Volume 4 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2014 > 4(3): 245-270

Next Article Previous Article

Mohamed Badr Benboubker, Hassane Hjiaj, Stanislas Ouaro. ENTROPY SOLUTIONS TO NONLINEAR ELLIPTIC ANISOTROPIC PROBLEM WITH VARIABLE EXPONENT[J]. Journal of Applied Analysis & Computation, 2014, 4(3): 245-270. doi: 10.11948/2014012

Citation:

Mohamed Badr Benboubker, Hassane Hjiaj, Stanislas Ouaro. ENTROPY SOLUTIONS TO NONLINEAR ELLIPTIC ANISOTROPIC PROBLEM WITH VARIABLE EXPONENT[J]. Journal of Applied Analysis & Computation, 2014, 4(3): 245-270. doi: 10.11948/2014012

ENTROPY SOLUTIONS TO NONLINEAR ELLIPTIC ANISOTROPIC PROBLEM WITH VARIABLE EXPONENT

1 Equipe de recherche SIGL, Ecole National des Sciences Appliquées, Université Abdelmalek Essaadi, BP 2222 M'hannech Tétouan, Maroc;
2 Laboratoire d'Analyse Mathématique et Applications(LAMA), Faculté des Sciences Dhar El Mahraz, Université Sidi Mohamed Ben Abdellah, BP 1796 Atlas Fès, Maroc;
3 LA boratoire de Mathématiques et Informatique(LAMI), UFR. Sciences Exactes et Appliquées, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

Abstract

Abstract

In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type -div(a(x, u, ∇u)) + g(x, u, ∇u) +|u|^p0(x)-2u=f -divϕ(u), in Ω, where -div(a(x, u, ∇u)) is a Leray-Lions operator, ϕ ∈ C⁰(IR, IR^N). The function g(x, u, ∇u) is a nonlinear lower order term with natural growth with respect to|∇u|, satisfying the sign condition and the datum f belongs to L¹(Ω).
- Anisotropic Sobolev spaces /
- variable exponent /
- nonlinear elliptic problem /
- entropy solutions
MSC: 35J20;35J25;35D30;35B38;35J60

FullText(HTML)

References

[1]	E. Azroul, A. Barbara, M.B. Benboubker and S. Ouaro, Renormalized solutions for a p(x)-Laplacian equation with Neumann nonhomogeneous boundary conditions and L¹-data, An. Univ. Craiova Ser. Mat. Inform., 40(1) (2013), 9-22. Google Scholar
[2]	E. Azroul, H. Hjiaj and A. Touzani, Existence and regularity of entropy solutions for strongly nonlinear p(x)-elliptic equations, Electronic J. Diff. Equ., 68(2013), 1-27. Google Scholar
[3]	H. Beiro da Veiga, On nonlinear potential theory, and regular boundary points, for the p-Laplacian in N space variables, Adv. Nonlinear Anal., 3(2014), 45-67. Google Scholar
[4]	M.B. Benboubker, E. Azroul and A. Barbara, Quasilinear elliptic problems with nonstandard growths, Electronic J. Diff. Equ., 62(2011), 1-16. Google Scholar
[5]	M. Bendahmane, M. Chrif and S. El Manouni, An approximation result in generalized anisotropic Sobolev spaces and application, Z. Anal. Anwend., 30(3) (2011), 341-353. Google Scholar
[6]	B.K. Bonzi, S. Ouaro and F D. Zongo, Entropy solutions for nonlinear elliptic anisotropic homogeneous Neumann problem, Int. J. Differ. Equ., 14(2013). Google Scholar
[7]	B.K. Bonzi, S. Ouaro and F D. Zongo, Entropy solutions to nonlinear elliptic anisotropic problems with Robin type boundary conditions, Matematiche, 68(2) (2013), 53-76. Google Scholar
[8]	M.M. Boureanu and V.D. Radulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. TMA, 75(12) (2012), 4471-4482. Google Scholar
[9]	A.C. Cavalheiro, Existence results for Dirichlet problems with degenerated pLaplacian, Opuscula Math., 33(3) (2013), 439-453. Google Scholar
[10]	L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011. Google Scholar
[11]	X.L. Fan and D. Zhao, On the generalised Orlicz-Sobolev space W^k,p(x)(Ω), J. Gansu Educ. College, 12(1) (1998), 1-6. Google Scholar
[12]	P. Harjulehto and P. Hästö, Sobolev inequalities for variable exponents attaining the values 1 and n, Publ. Mat., 52(2) (2008), 347-363. Google Scholar
[13]	E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-verlng, Berlin Heidelberg New York, 1965. Google Scholar
[14]	B. Koné, S. Ouaro and S. Traoré,Weak solutions for anisotropic nonlinear elliptic equations with variable exponent, Electron. J. Diff. Equ., 144(2009), 1-11. Google Scholar
[15]	B. Koné, S. Ouaro and S. Soma, Weak solutions for anisotropic nonlinear elliptic problems with variable exponent and measure data, Int. J. Evol. Equ., 5(3) (2011), 327-350. Google Scholar
[16]	J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod et Gauthiers-Villars, Paris 1969. Google Scholar
[17]	M. Mihailescu, P. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340(2008), 687-698. Google Scholar
[18]	S. Ouaro, Well-posedness results for anisotropic nonlinear elliptic equations with variable exponent and L¹-data, Cubo, 12(1) (2010), 133-148. Google Scholar
[19]	D. Zhao, W.J. Qiang and X.L. Fan, On generalized Orlicz spaces L^p(x)(Ω), J. Gansu Sci., 9(2) (1997), 1-7. Google Scholar
[20]	V.V. Zhikov, On Lavrentiev's phenomenon, Russian J. Math. Phys., 3(1995), 249-269. Google Scholar
[21]	V.V. Zhikov, On some variational problem, Russian J. Math. Phys., 5(1997), 105-116. Google Scholar