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