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We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on $\Omega$. We prove the global $C^{1,\alpha }$ regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the $C^{1,\alpha }$ regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case.

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.