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 the generalized Gehring lemmas under some general conditions on the exponent p (x) and establish a modified version of the Sobolev-Poincaré inequality to deal with the Sobolev norms with variable exponents; these results 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 known results obtained within the logarithmic Zhikov framework.
Chen et al. (Thu,) studied this question.