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.
Highlights
Partial differential equations involving the p(x)-Laplacian arise, for instance, in nonlinear elasticity, fluid mechanics, non-Newtonian fluids and image processing
One of the approaches to study the existence of solutions of elliptic partial differential equations is the sub-supersolution method
The main aim of this paper is to prove the existence of a weak positive solution for (1.2) via the sub-supersolution method
Summary
Partial differential equations involving the p(x)-Laplacian arise, for instance, in nonlinear elasticity, fluid mechanics, non-Newtonian fluids and image processing. One of the approaches to study the existence of solutions of elliptic partial differential equations is the sub-supersolution method. The lack of the existence of the first eigenvalue implies a considerable difficulty when dealing with boundary value problems involving the p(x)-Laplacian by using the subsupersolution method. Papers that consider such problems by using the mentioned method are rare in the literature. We present some preliminaries to construct a function space where the solution of (1.2) makes sense
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