Abstract
We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the p(x)-Laplacian operator with Dirichlet boundary condition: \t\t\t−Δp(x)u+V(x)|u|q(x)−2u=f(x,u)in Ω,u=0 on ∂Ω,\\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}$$ -\\Delta _{p(x)}u+V(x) \\vert u \\vert ^{q(x)-2}u =f(x,u)\\quad \\text{in }\\varOmega , u=0 \\text{ on }\\partial \\varOmega , $$\\end{document} where Ω is a smooth bounded domain in mathbb{R}^{N}, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, f(x,t) is a Carathéodory function satisfying some growth conditions. Depending on the assumptions, the solutions set may consist of a bounded infinite sequence of solutions or a unique one. Our technique is based on a symmetric version of the mountain pass theorem.
Highlights
1 Introduction In this work, we study the existence of solutions for the following nonlinear Dirichlet problem involving the p(x)-Laplacian operator:
Let us recall that the p(x)-Laplacian operator p(x) is defined by p(x)u = div |∇u|p(x)–2∇u
One can name for instance electrorheological fluids [30, 32, 36], elastic mechanics, flows in porous media and image processing [11], curl systems emanating from electromagnetism [4, 7]
Summary
In [8, 9], the authors considered problem (1.1) with V bounded and p(·) = q(·) and proved the existence of nonnegative solutions using a mountain pass theorem. Proposition 2.14 Suppose that f : Ω × R → R is a Carathéodory function satisfying the following growth condition: α(x)
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