Existence of infinitely many solutions for double phase problem with sign-changing potential

In this paper, we investigate the existence of infinitely many solutions for the following double phase problem -div(|∇u|p-2∇u+a(x)|∇u|q-2∇u)=f(x,u),inΩ,u=0,on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\mathrm{div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)= f(x,u),&{}\hbox {in }\;\Omega , \\ u=0, &{}\hbox {on }\;\partial \Omega , \end{array} \right. \end{aligned}$$\end{document}where N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document} and 1<p<q<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<q<N$$\end{document}. Based on a direct sum decomposition of a space W01,H(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_0^{1,H}(\Omega )$$\end{document}, we prove that the above problem possesses multiple solutions under mild assumptions on a and f. The primitive of the nonlinearity f is of super-q growth near infinity in u and allowed to be sign-changing. Furthermore, our assumptions are suitable and different from those studied previously.