Abstract
For a class of second-order anisotropic elliptic equations with variable nonlinearity indices and summable right-hand sides, we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable exponents.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Aharouch, J. Bennouna, and A. Touzani, “Existence of renormalized solution of some elliptic problems in Orlicz spaces,” Rev. Mat. Complut., 22, No. 1, 91–110 (2009).
A. Alvino, L. Boccardo, V. Ferone, L. Orsina, and G. Trombetti, “Existence results for nonlinear elliptic equations with degenerate coercivity,” Ann. Mat. Pura Appl. (4), 182, No. 1, 53–79 (2003).
E. Azroul, H. Hjiaj, and A. Touzani, “Existence and regularity of entropy solutions for strongly nonlinear p(x)-elliptic equations,” Electron. J. Differ. Equ., 2013, No. 68, 1–27 (2013).
M. B. Benboubker, E. Azroul, and A. Barbara, “Quasilinear elliptic problems with nonstandard growths,” Electron. J. Differ. Equ., 2011, No. 62, 1–16 (2011).
M. B. Benboubker, H. Chrayteh, M. El Moumni, and H. Hjiaj, “Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data,” Acta Math. Sin. (Engl. Ser.), 31, No. 1, 151–169 (2015).
M. B. Benboubker, H. Hjiaj, and S. Ouaro, “Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent,” J. Appl. Anal. Comput., 4, No. 3, 245–270 (2014).
M. Bendahmane and K. Karlsen, “Nonlinear anisotropic elliptic and parabolic equations in ℝN with advection and lower order terms and locally integrable data,” Potential Anal., 22, No. 3, 207–227 (2005).
M. Bendahmane and P. Wittbold, “Renormalized solutions for nonlinear elliptic equations with variable exponents and L1-data,” Nonlinear Anal., 70, No. 2, 567–583 (2009).
Ph. Benilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre, and J. L. Vazquez, “An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22, No. 2, 241–273 (1995).
A. Benkirane and J. Bennouna, “Existence of entropy solutions for some elliptic problems involving derivatives of nonlinear terms in Orlicz spaces,” Abstr. Appl. Anal., 7, No. 2, 85–102 (2002).
M. F. Bidaut-Veron, “Removable singularities and existence for a quasilinear equation with absorption or source term and measure data,” Adv. Nonlinear Stud., 3, 25–63 (2003).
L. Boccardo and Th. Gallouët, “Nonlinear elliptic equations with right-hand side measures,” Commun. Part. Differ. Equ., 17, No. 3-4, 641–655 (1992).
L. Boccardo, Th. Gallouët, and P. Marcellini, “Anisotropic equations in L1,” Differ. Integral Equ., 9, No. 1, 209–212 (1996).
L. Boccardo, T. Gallouët, and J. L. Vazquez, “Nonlinear elliptic equations in RN without growth restrictions on the data,” J. Differ. Equ., 105, No. 2, 334–363 (1993).
B. K. Bonzi and S. Ouaro, “Entropy solutions for a doubly nonlinear elliptic problem with variable exponent,” J. Math. Anal. Appl., 370, 392–405 (2010).
H. Brezis, “Semilinear equations in ℝN without condition at infinity,” Appl. Math. Optim., 12, No. 3, 271–282 (1984).
L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin–Heidelberg (2011).
A. El Hachimi and A. Jamea, “Uniqueness result of entropy solution to nonlinear neumann problems with variable exponent and L1-data,” J. Nonlinear Evol. Equ. Appl., 2017, No. 2, 13–25 (2017).
X. Fan, “Anisotropic variable exponent Sobolev spaces and p(x)-Laplacian equations,” Complex Var. Elliptic Equ., 56, No. 7–9, 623–642 (2011).
P. Gwiazda, P. Wittbold, A. Wróblewska, and A. Zimmermann, “Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces,” J. Differ. Equ., 253, 635–666 (2012).
T. C. Halsey, “Electrorheological fluids,” Science, 258, No. 5083, 761–766 (1992).
A. A. Kovalevskiy, “A priori properties of solutions of nonlinear equations with degenerate coercitivity and L1-data,” Sovrem. Mat. Fundam. Napravl., 16, 47–67 (2006).
A. A. Kovalevskiy, “On convergence of functions from Sobolev space satisfying special integral estimates,” Ukr. Mat. Zh., 58, No. 2, 168–183 (2006).
L. M. Kozhevnikova, “On entropy solution of an elliptic problem in anisotropic Sobolev–Orlich spaces,” Zh. Vychisl. Mat. Mat. Fiz., 57, No. 3, 429–447 (2017).
L. M. Kozhevnikova, “Existence of entropy solutions of an elliptic problem in anisotropic Sobolev–Orlich spaces,” Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., 139, 15–38 (2017).
L. M. Kozhevnikova and A. Sh. Kamaletdinov, “Existence of solutions of anisotropic elliptic equations with variable nonlinearity index in unbounded domains,” Vestn. Volgograd. Gos. Univ. Ser. 1. Mat. Fiz., No. 5(36), 29–41 (2016).
S. N. Kruzhkov, “First-order quasilinear equations with multiple independent variables,” Mat. Sb., 81, No. 123, 228–255 (1970).
S. Ouaro, “Well-posedness results for anisotropic nonlinear elliptic equations with variable exponent and L1-data,” Cubo, 12, No. 1, 133–148 (2010).
M. Sancho’n and J. M. Urbano, “Entropy solutions for the p(x)-Laplace equation,” Trans. Am. Math. Soc., 361, No. 12, 6387–6405 (2009).
V. V. Zhikov, “On variational problems and nonlinear elliptic equations with nonstandard growth conditions,” Probl. Mat. Anal., 54, 23–112 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 3, Differential and Functional Differential Equations, 2017.
Rights and permissions
About this article
Cite this article
Kozhevnikova, L.M. On Entropy Solutions of Anisotropic Elliptic Equations with Variable Nonlinearity Indices in Unbounded Domains. J Math Sci 253, 692–709 (2021). https://doi.org/10.1007/s10958-021-05262-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05262-0