Let be a bounded open set of , , , . Consider the nonlinear elliptic equations
where is a Carathéodory function satisfying the classical Leray-Lions hypotheses. Assume that weakly in , strongly in and a.e. in , and strongly in . Moreover, assume that is bounded in the space of Radon measures.
In the present paper, the authors prove that strongly in for any . This implies that, for a suitable subsequence , a.e. in (cf. the title of the paper) and, moreover, that it is allowed to pass to the limit in (1) such that in .
Besides, under the stronger hypotheses for some (a.e. , and , arbitrary) and weakly in , they show that, for any fixed , the truncation of at height satisfies strongly in . Under suitably modified assumptions, corresponding results are obtained also in the parabolic case, i.e., when (1) is replaced by