Existence of weak solutions for a $p(x)$-Laplacian equation via topological degree

We consider the $p(x)$-Laplacian equation with a Dirichlet boundary value condition $$ \begin{cases} -\Delta_{p(x)}(u)+|u|^{p(x)-2}u= g(x,u,\nabla u), &x\in\Omega,\\ u=0, &x\in\partial\Omega. \end{cases} $$ Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions, the existence of weak solutions for this equation.