Entropy Solutions to a Strongly Nonlinear Unilateral Obstacle Elliptic Problem with Orlicz Growth

In this paper , we investigate an existence result of solutions for the nonlinear elliptic unilateral problem u≥ψa.e. inΩ,Tk(u)∈W01LM(Ω),∫ΩA(x,u,∇u)∇Tk(u-φ)dx+∫ΩΦ(x,u)∇Tk(u-φ)dx≤∫ΩfTk(u-φ)dx,∀φ∈L∞(Ω)∩(Kψ={u∈W01LM(Ω):u≥ψa.e.inΩ}),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} u\ge \psi \text { a.e. in }\Omega , T_k(u)\in W_{0}^{1}L_{M}(\Omega ),\\ \displaystyle \int _{\Omega }{\mathcal {A}}(x,u,\nabla u)\nabla T_k(u-\varphi )\>\mathrm{d}x+\displaystyle \int _{\Omega }\Phi (x,u)\nabla T_k(u-\varphi )\>\mathrm{d}x\\ \le \displaystyle \int _{\Omega }fT_k(u-\varphi )\>\mathrm{d}x,\\ \quad \forall \varphi \in L^{\infty }(\Omega )\cap \Big ({\mathbf {K}}_{\psi }= \Big \{u\in W_{0}^{1}L_{M}(\Omega ):u\ge \psi \text{ a.e. } \text{ in } \Omega \Big \}\Big ), \end{array}\right. \end{aligned}$$\end{document}where the lower-order term Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} verifies a generalized natural growth condition described by a suitable N-function M and the data f is an element of L1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1(\Omega )$$\end{document}. No restriction is assumed neither on M nor on its conjugate M¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{M}}$$\end{document}.