Dirichlet boundary value problem related to the $p(x)-$Laplacian with discontinuous nonlinearity

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$-div(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x)-2}\mathrm{\nabla }u)+u\in -[\underset{‾}{g}(x,u),\bar{g}(x,u)],$$ by using the degree theory after turning the problem into a Hammerstein equation. The right hand side $g$ is a possibly discontinuous function in the second variable satisfying some non-standard growth conditions…