Anisotropic (p, q)-Equations with Asymmetric Reaction Term

We consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian (double phase problem) and with a reaction term which exhibits asymmetric behavior as x→±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow \pm \infty $$\end{document}. Using variational tools, truncation, and comparison techniques and critical groups, we prove a multiplicity theorem producing four nontrivial solutions all with sign information and ordered.