Non-linear elliptic unilateral problems with two lower-order terms in Orlicz spaces

摘要

Abstract This paper is devoted to the existence of solutions in the context of Orlicz spaces for strongly nonlinear elliptic unilateral problems with a Leray-Lions operator A acting from W01LM(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$W^{1}_{0}L_{M}(Omega )$$end{document} to its dual for N-function M, and two lower-order terms g(x,u,∇u)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$g(x, u, nabla u)$$end{document} and H(x,∇u)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H(x,nabla u)$$end{document}. It is assumed that the non-linear term g satisfies a growth condition on ∇udocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$nabla u$$end{document} and a sign condition on u, while the convection term H is only growing at most as M-1M(|∇u|)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$M^{-1}M(|nabla u|)$$end{document}. The source term f belongs to 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}.