On some general multiplying solutions results of a Robin problem

Abstract

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( \omega _{1}(x)\left\vert \nabla u\right\vert^{p(x)-2}\nabla u\right) =\lambda \omega _{2}(x)f(x,u), & x\in \Omega \\ \omega _{1}(x)\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{ \partial \upsilon }+\beta (x)\left\vert u\right\vert ^{p(x)-2}u=0, & x\in \partial \Omega , \end{array} \right. \end{equation*} in $W_{\omega _{1},\omega _{2}}^{1,p(.)}\left( \Omega \right) $ under some appropriate conditions.