Higher integrability for p(x)-Laplacian equations with drift terms and non-logarithmic Zhikov's conditions

In this paper, we study a higher integrability of gradient for the weak solutions of $p(x)$-Laplacian equations involving a drift term. We present two different versions of Gehring's generalized lemmas under some general conditions on the exponent $p(x)$ and establish a modified version of Sobolev-Poincaré inequality to deal with the Sobolev norms with variable exponents; these results will optimize some known Zhikov's conditions for higher integrability. We also give some sufficient conditions on the drift term which will guarantee the solvability of the Dirichlet problem and the higher integrability of gradient for the weak solutions. Our results generalize the result under the logarithmic Zhikov's condition.