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 ${\mathbb{R}}^n$, which are obtained through the use of the iteration-covering method. More precisely, a global Calderón-Zygmund type result

$$|f{|}^{p_1}+a(x)|f{|}^{p_2}\in L^s({\mathbb{R}}^n)\Rightarrow |Du{|}^{p_1}+a(x)|Du{|}^{p_2}\in L^s({\mathbb{R}}^n)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}s>1$$

is established for the weak solutions of

$$-\mathrm{d}\mathrm{i}\mathrm{v}A(x,Du)=-\mathrm{d}\mathrm{i}\mathrm{v}F(x,f)\mathrm{i}\mathrm{n}{\mathbb{R}}^n,$$

which are modeled on

$$-\mathrm{d}\mathrm{i}\mathrm{v}(|Du{|}^{p_1-2}Du+a(x)|Du{|}^{p_2-2}Du)=-\mathrm{d}\mathrm{i}\mathrm{v}(|f{|}^{p_1-2}f+a(x)|f{|}^{p_2-2}f),$$

where $0\le a(\cdot )\in C^{0,\alpha }({\mathbb{R}}^n),\alpha \in (0,1]$ and $1<p_1<p_2<p_1+\frac{\alpha p_1}{n}$.