The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

Abstract

In this paper, a nonhomogeneous elliptic equation of the form −A(x,|u|Lr(x))div(a(|∇u|p(x))|∇u|p(x)−2∇u)=f(x,u)|∇u|Lq(x)α(x)+g(x,u)|∇u|Ls(x)γ(x)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& - \\mathcal{A}\\bigl(x, \\vert u \\vert _{L^{r(x)}}\\bigr) \\operatorname{div}\\bigl(a\\bigl( \\vert \\nabla u \\vert ^{p(x)}\\bigr) \\vert \\nabla u \\vert ^{p(x)-2} \\nabla u\\bigr) \\\\& \\quad =f(x, u) \\vert \\nabla u \\vert ^{\\alpha (x)}_{L^{q(x)}}+g(x, u) \\vert \\nabla u \\vert ^{ \\gamma (x)}_{L^{s(x)}} \\end{aligned}$$ \\end{document} on a bounded domain Ω in {mathbb{R}}^{N} (N >1) with C^{2} boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.