Gradient estimates of very weak solutions to general quasilinear elliptic equations (Q2170659)

The authors study existence and regularity of very weak solutions of nonlinear elliptic equations with nonstandard growth of the form \[ \operatorname{div} A (x,Du)=\operatorname{div} \left(\frac{\varphi'|\mathbf{f}|}{|\mathbf{f}|} \right)\text{ in }\Omega, \] where \(\Omega\subset \mathbb R^n\), \(n\geq 2\) is a bounded domain, \(\varphi\) is a given Young function, and \(A(x,\xi)\) is a Caratheéodory map satisfying suitable growth and monotonicity conditions, that make \(A\) a natural generalization of the \(p\)-Laplace equation. It is obtained a Caldéron-Zygmund-type estimate for the gradient of the solution in Orlicz-Sobolev spaces, that implies \(\varphi(|Du|) \in L^q_{\mathrm{loc}}(\Omega)\) if \(\varphi(|\mathbf{f}|)\in L^q_{\mathrm{loc}}(\Omega)\) with \(q\in[1-\delta,1+\delta]\) for a small enough \(\delta\). The authors prove also an existence of very weak solutions to such nonlinear problems.