  By applying Mountain Pass Lemma, Ekeland's and Ricceri's variational
principle, Fountain Theorem, we prove the existence and multiplicity of
solutions for the following Robin problem \begin{equation*} \left\{
\begin{array}{cc} -\text{div}\left( a(x)\left\vert \nabla u\right\vert
^{p(x)-2}\nabla u\right) =\lambda b(x)\left\vert u\right\vert ^{q(x)-2}u, &
x\in \Omega \\ a(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*} under some appropriate
conditions in the space $W_{a,b}^{1,p(.)}\left( \Omega \right).$
