Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent

Elliptic equations of p(x)-Laplacian type are investigated. There is a well-known logarithmic condition on the modulus of continuity of the nonlinearity exponent p(x), which ensures that a Laplacian with variable order of nonlinearity inherits many properties of the usual p-Laplacian of constant order. One of these is the so-called improved integrability of the gradient of the solution. It is proved in this paper that this property holds also under a slightly more general condition on the exponent p(x), although then the improvement of integrability is logarithmic rather than power-like. The method put forward is based on a new generalization of Gehring's lemma, which relies upon the reverse Holder inequality "with increased support and exponent on the right-hand side". A counterexample is constructed that reveals the extent to which the condition on the modulus of continuity obtained is sharp.