On some general multiplying solutions results of a Robin problem

By applying Ricceri's variational principle, we demonstrate the existence of solutions for the following Robin problem \begin{equation*}\left\{ \begin{array}{cc}-\func{div}\left( ω_{1}(x)\left\vert \nabla u\right\vert^{p(x)-2}\nabla u\right) =λω_{2}(x)f(x,u), & x\in Ω\\ ω_{1}(x)\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{ \partial \upsilon }+β(x)\left\vert u\right\vert ^{p(x)-2}u=0, & x\in \partial Ω, \end{array} \right. \end{equation*} in $W_{ω_{1},ω_{2}}^{1,p(.)}\left( Ω\right) $ under some appropriate conditions.