Existence of weak solutions for the anisotropic $p(x)$-Laplacian via degree theory

In this paper, we consider Dirichlet boundary value problem involving the anisotropic $p(x)$-Laplacian, where $p(x)= (p_1(x), ..., p_n(x))$, with $p_i(x)> 1$ in $\overline{\Omega}$. Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions on the data, the existence of weak solutions for the given problem. An important contribution is that we are considering the degenerate and the singular cases in the discussion. Finally, according to the compact embedding for anisotropic Sobolev spaces, we point out that the considered boundaru value problem may be critical in some region of $\Omega$.