Existence of weak solutions for elliptic Dirichlet problems with variable exponent

summary:This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type $$ \begin{cases} -{\rm div} a(x, u, \nabla u)+b(x, u, \nabla u)=0 &\text {in} \ \Omega ,\\ u=0 &\text {on} \ \partial \Omega , \end{cases} $$ where $\Omega $ is a bounded domain of $\mathbb R^n$, $n\ge 2$. In particular, we do not require strict monotonicity of the principal part $a(x,z,\cdot )$, while the approach is based on the variational method and results of the variable exponent function spaces