General quasilinear problems involving $p(x)$-Laplacian with Robin boundary condition

This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving $p(x)$-Laplace type equation, namely
$$\begin{equation*}\label{E11} \left\{\begin{array}{lll} -\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\ n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega. \end{array}\right. \end{equation*}$$
Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.