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On very weak solutions of a class of nonlinear elliptic systems. (English) Zbl 1119.35016
The paper is devoted to the study of a very weak solution of the equation $$\text{div}\;A(x,u,Du)=B(x,u,Du)$$, where $$A$$, $$B$$ grow in the gradient like $$t^{p-1}$$ and $$B(x,u,Du)$$ is not in divergence form. The author prove that a very weak solution $$u\in W^{1,r}$$ of this equation belongs to $$W^{1,p}$$.

##### MSC:
 35J50 Variational methods for elliptic systems 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35D05 Existence of generalized solutions of PDE (MSC2000) 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Keywords:
nonlinear elliptic system; maximal operator
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