Abstract
Here, we consider the following elliptic problem with variable components: −a(x)Δp(x)u−b(x)Δq(x)u+u|u|s−2|x|s=λf(x,u),\\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}$$ -a(x)\\Delta _{p(x)}u - b(x) \\Delta _{q(x)}u+ \\frac{u \\vert u \\vert ^{s-2}}{|x|^{s}}= \\lambda f(x,u), $$\\end{document} with Dirichlet boundary condition in a bounded domain in mathbb{R}^{N} with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.
Highlights
1 Introduction The quasilinear operator (p, q)-Laplacian has been used to model steady-state solutions of reaction–diffusion problems arising in biophysics, plasma physics, and in the study of chemical reactions
The differential operator p + q is known as the (p, q)-Laplacian operator, if p = q, where j, j > 1 denotes the j-Laplacian defined by ju := div(|∇u|j–2∇u)
Our main interest in this work is to prove the existence of a weak solution of the weighted (p(x), q(x))-Laplacian problem
Summary
The quasilinear operator (p, q)-Laplacian has been used to model steady-state solutions of reaction–diffusion problems arising in biophysics, plasma physics, and in the study of chemical reactions. Our main interest in this work is to prove the existence of a weak solution of the weighted (p(x), q(x))-Laplacian problem (2021) 2021:80 where ⊂ RN is a bounded domain with a smooth boundary, a, b ∈ L∞( ) are positive functions with a(x) ≥ 1 a.e. on , λ > 0 is a real parameter, r(x)u = div(|∇u|r(x)–2∇u) denotes r(x)-Laplacian operator, for r ∈ {p, q}, where p, q ∈ C+( ̄ ), 1 < s < q(x) < p(x) < ∞ a.e. on and f : × R → R is a Carathéodory function satisfying the following growth condition: (f1) f (x, t) ≤ α + β|t|h(x)–1 for all (x, t) ∈ × R where a1, a2 are two nonnegative constants, h ∈ C+( ̄ ) with h(x) < p∗(x) a.e. in and
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