Gradient estimates for the double phase problems in the whole space

This paper presents Calder on-Zygmund estimates for the weak solutions of a class of nonuniformly elliptic equations inRn, which are obtained through the use of the iteration-covering method. More precisely, a global Calder on-Zygmund type result |f|(p1)+a(x)|f|(p2)is an element of(LRn)-R-s()double right arrow|Du|(p1)+a(x)|Du|(p2)is an element of L-s(R-n) for any s>1 is established for the weak solutions of -divA(x,Du)=-divF(x,f) in R-n, which are modeled on, -div(|Du|(p1-2)Du+a(x)|Du|(p2-2)Du)=-div(|f(|p1-2)f+a(x)|f|(p2-2)f),where 0 <= a(<middle dot>)is an element of C0,alpha(R-n), alpha is an element of(0,1] and 1<p1<p2<p1+alpha p1/n.