Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent
Vasilii Vasil'evich Zhikov, Svetlana Evgenievna Pastukhova · Sbornik Mathematics · 2008
Elliptic equations of -Laplacian type are investigated. There is a?well-known logarithmic condition on the modulus of continuity of the nonlinearity exponent , which ensures that a?Laplacian with variable order of nonlinearity inherits many properties of the usual -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 , 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 H?lder 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. Bibliography: 28 titles.