Several versions of the compensated compactness principle

The convergence of the product of a solenoidal vector $w_\varepsilon$ and a gradient $\nabla u_\varepsilon$ in $L^1(\Omega)$ (where $\Omega$ is a region in $\mathbb R^d$) is investigated in the case when the factors converge weakly in the spaces $L^\gamma(\Omega)^d$ and $L^\alpha(\Omega)^d$, respectively, with $1/\gamma+1/\alpha>1$, which means that the main assumption of the classical $div$-$curl$ lemma fails. Nevertheless, the same convergence (in the sense of distributions in $\Omega$)
$$ \lim_{\varepsilon\to0}w_\varepsilon\cdot\nabla u_\varepsilon =\lim_{\varepsilon\to0}w_\varepsilon\cdot\lim_{\varepsilon\to0} \nabla u_\varepsilon=w\cdot\nabla u $$
as in the framework of the $div$-$curl$ lemma, survives under certain additional assumptions.
The new versions of the compensated compactness principle proved in the paper can be used in homogenization and in the theory of $G$-convergence of monotone operators with non-standard coercivity and growth properties, for instance, some degenerate operators.
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