Nonlinear elliptic equations with measure data in Orlicz spaces
Abstract
UDC 517.5
In this article, we study the existence result of the unilateral problem
\begin{gather*}
Au-\mbox{div} (\Phi(x,u))+H(x,u,\nabla u)=\mu,
\end{gather*}
where $Au = -\mbox{div}(a(x,u,\nabla u))$ is a Leray–Lions operator defined on Sobolev–Orlicz space $D(A)\subset W_{0}^{1}L_{M}(\Omega),$ $\mu \in L^{1}(\Omega)+W^{-1}E_{\overline{M}}(\Omega),$ where $M$ and $\overline{M}$ are two complementary $N$-functions, the first and the second lower terms $\Phi$ and $H$ satisfies only the growth condition and any sign condition is assumed and $u\geq \zeta,$ where $\zeta$ is a measurable function.
References
R. A. Adams, Sobolev spaces, Acad. Press, New York (1975).
A. Aberqi, J. Bennouna, M. Elmassoudi, M. Hammoumi, Existence and uniqueness of a renormalized solution of parabolic problems in Orlicz spaces, Monatsh. Math. 189, № 2, 195--219, (2019); https://doi.org/10.1007/s00605-018-01260-8 DOI: https://doi.org/10.1007/s00605-018-01260-8
A. Aberqi, J. Bennouna, M. Mekkour, H. Redwane, Nonlinear parabolic inequality with lower order terms, Appl. Anal. 96, № 12, 2102--2117, (2017); https://doi.org/10.1080/0036811.2016.1205186 DOI: https://doi.org/10.1080/00036811.2016.1205186
A. Aberqi, J. Bennouna, H. Redwane, Nonlinear parabolic Problems with lower order terms and measure data, Thai J. Math., 14, № 1, 115 – 130 (2016).
L. Aharouch, E. Azroul, M. Rhoudaf, Nonlinear unilateral problems in Orlicz spaces, Appl. Math., 33, №2, 217 – 241(2006), https://doi.org/10.4064/am33-2-6 DOI: https://doi.org/10.4064/am33-2-6
E. Azroul, H. Redwane, M. Rhoudaf, Existence of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces, Port. Math., 66, № 1, 29 – 63 (2009), https://doi.org/10.4171/PM/1829 DOI: https://doi.org/10.4171/PM/1829
A. Benkirane, J. Bennouna, Existence and uniqueness of solution of unilateral problems with L1 -data in Orlicz spacesExistence of entropy solutions for some nonlinear problems in Orlicz spaces, Abstr. and Appl. Anal., 7, № 2 , 85 – 102 (2002), https://doi.org/10.1155/S1085337502000751 DOI: https://doi.org/10.1155/S1085337502000751
A. Benkirane, A. Elmahi, Almost everywhere convergence of the gradients of solutions to elliptic equations problems in Orlicz spaces and application, Nonlinear Anal., 28, № 11, 1769 – 1784 (1997); https://doi.org/10.1016/S0362-546X(96)00017-X DOI: https://doi.org/10.1016/S0362-546X(96)00017-X
P. B´enilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vázquez, An $L^1$ -theory of existence anduniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Super Pisa, 22, № 2, 241 – 273 (1995).
L. Boccardo, T. Gallouet, L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Funct. Anal., 87, 149 – 169 (1989).
L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. Henri Poimcar´e C, 13, № 5, 539 – 551 (1996); https://doi.org/10.1016/S0294-1449(16)30113-5 DOI: https://doi.org/10.1016/s0294-1449(16)30113-5
M. Elmassoudi, A. Aberqi, J. Bennouna, Nonlinear parabolic problem with lower order terms in Musielack – Orlicz spaces, ASTES J., 2, № 5, 109 – 123 (2017); https://doi.org/10.25046/aj020518 DOI: https://doi.org/10.25046/aj020518
J. P. Gossez, V. Mustonen, Variational inequalities in Orlicz – Sobolev spaces, Nonlinear Anal., 11, № 3, 379 – 492 (1987); https://doi.org/10.1016/0362-546X(87)90053-8 DOI: https://doi.org/10.1016/0362-546X(87)90053-8
J. P. Gossez, Some approximation properties in Orlicz – Sobolev spaces, Stud. Math., 74, № 1, 17 – 24 (1982), https://doi.org/10.4064/sm-74-1-17-24 DOI: https://doi.org/10.4064/sm-74-1-17-24
J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Super Pisa, 18, 189 – 258, (1964).
Copyright (c) 2021 Mhamed Elmassoudi
This work is licensed under a Creative Commons Attribution 4.0 International License.
