Gradient estimates for the double phase problems in the whole space.

This paper presents Calderón-Zygmund estimates for the weak solutions of a class of nonuniformly elliptic equations in R n , which are obtained through the use of the iteration-covering method. More precisely, a global Calderón-Zygmund type result | f | p 1 + a (x) | f | p 2 ∈ L s (R n) ⇒ | D u | p 1 + a (x) | D u | p 2 ∈ L s (R n) f o r a n y s > 1 is established for the weak solutions of − d i v A (x , D u) = − d i v F (x , f) i n R n , which are modeled on − d i v (| D u | p 1 − 2 D u + a (x) | D u | p 2 − 2 D u) = − d i v (| f | p 1 − 2 f + a (x) | f | p 2 − 2 f) , where 0 ≤ a (⋅) ∈ C 0 , α (R n) , α ∈ (0 , 1 ] and 1 < p 1 < p 2 < p 1 + α p 1 n . This paper presents Calderón-Zygmund estimates for the weak solutions of a class of nonuniformly elliptic equations in R n , which are obtained through the use of the iteration-covering method. More precisely, a global Calderón-Zygmund type result | f | p 1 + a (x) | f | p 2 ∈ L s (R n) ⇒ | D u | p 1 + a (x) | D u | p 2 ∈ L s (R n) f o r a n y s > 1 is established for the weak solutions of − d i v A (x , D u) = − d i v F (x , f) i n R n , which are modeled on − d i v (| D u | p 1 − 2 D u + a (x) | D u | p 2 − 2 D u) = − d i v (| f | p 1 − 2 f + a (x) | f | p 2 − 2 f) , where 0 ≤ a (⋅) ∈ C 0 , α (R n) , α ∈ (0 , 1 ] and 1 < p 1 < p 2 < p 1 + α p 1 n . [ABSTRACT FROM AUTHOR]