Regularity for solutions to nonlinear elliptic equations. (English) Zbl 1299.35124

The authors consider elliptic equations modeled on \[ \text{div} \left [| \nabla u| ^{p-2}\nabla u+B(x)| u| ^{p-2}u\right ]=\text{div}\left (| F| ^{p-2}F\right ) \] in domains \(\Omega \subset \mathbb {R}^N\), with \(1<p<N\). The coefficient \(B(x)\) is assumed to belong to the Marcinkiewicz space \(L^{\frac {N}{p-1},\infty }\) so that it is also close in the norm of this space to a bounded function. The authors prove that \(F\in L^r_{\text{loc}}\) for some \(p<r<N\) implies that \(u\in L^{r^{\ast }}_{\text{loc}}\). Here, \(r^{\ast }\) is the Sobolev conjugate of \(r\). An example showing that the assumption on closeness to a bounded function is necessary for the results is also provided.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data