Almost all weak solutions of the weighted p(.)-biharmonic problem

We give a new compact embedding theorem, and use an equivalent norm to obtain some different solutions of the weighted p.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\left( .\right) $$\end{document}-biharmonic eigenvalue problem Δν(x)Δup(x)-2Δu=λω(x)uq(x)-2u,inΩu=Δu=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}$$\begin{aligned} \left\{ \begin{array}{cc} \Delta \left( \nu (x)\left| \Delta u\right| ^{p(x)-2}\Delta u\right) =\lambda \omega (x)\left| u\right| ^{q(x)-2}u, &{} \text {in }\Omega \\ u=\Delta u=0, &{} \text {on }\partial \Omega . \end{array} \right. \end{aligned}$$\end{document}Using Mountain Pass Theorem we show that the problem has a nontrivial weak solution. Moreover, we obtain infinite many pairs of solutions of the problem due to the Fountain Theorem.