Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb R^{N}$, ${{2^*}}=\frac{2N}{N-2}$, $a>0$, $S=\inf\big\{ \int_{\mathbb R^{N}}|\nabla u|^2 : u\in L^{{2^*}}(\mathbb R^{N}), \nabla u\in L^2(\mathbb R^{N}), \int_{\mathbb R^{N}}|u|^{{2^*}}$ $=1 \big\}$ and $||u||^2= |\nabla u|_{2}^2+a|u|_{2}^2$. We define ${{2^\flat}}=\frac{2N}{N-1}$, ${{2^\#}}=\frac{2(N-1)}{N-2}$ and consider $q$ such that ${{2^\flat}}\leq q\leq{{2^\#}}$. We also define $s=2-N+\frac{q}{{{2^*}}-q}$ and $t=\frac{2}{N-2}\cdot\frac{1}{{{2^*}}-q}$. We prove that there exists an $\alpha_{0}(q,a,\Omega)>0$ such that, for all $u\in H^1(\Omega)\setminus\{0\}$, $$ {\frac{S}{2^{\frac 2N}}} {{|u|_{{2^*}}^2}} \leq{||u||^2}+\alpha_{0} {\Big(\frac{{||u||}}{|u|_{{{2^*}}}^{{{2^*}}\!/2}}\Big)^{s}} {|u|_{q}^{qt}} , \tag*{(I)_{q}}$$ where the norms are over $\Omega$. Inequality $(I)_{{{2^\flat}}}$ is due to M. Zhu.

The longtime behaviour of solutions of a reaction-diffusion system with the nonlinearity rapidly oscillating in time ($f=f(t/{\varepsilon},u)$) is studied. It is proved that (under the natural assumptions) this behaviour can be described in terms of global (uniform) attractors $\mathcal A^{\varepsilon}$ of the corresponding dynamical process and that these attractors tend as ${\varepsilon}\to 0$ to the global attractor $\mathcal A^0$ of the averaged autonomous system. Moreover, we give a detailed description of the attractors $\mathcal A^{\varepsilon}$, ${\varepsilon}\ll1$, in the case where the averaged system possesses a global Liapunov function.

We study the Hamilton-Jacobi equation \[H(x,Du)=0 , \] where $H(x,p)$ is assumed to be measurable in $x$, quasiconvex and continuous in $p$. The notion of viscosity solution is adapted to the measurable setting making use of suitable measure--theoretic devices. We obtain integral representation formulae generalizing the ones valid for continuous equations, comparison principles and uniqueness results. We examine stability properties of the new definition and present two approximation procedures: the first one is based on a regularization of the Hamiltonian by mollification and in the second one the approximating sequence is made up by minimizers of certain variational integrals.