We give a priori estimates for nonlinear partial differential equations in which the ellipticity bounds degenerate. The so-called distortion function which measures the degree of degeneracy of ellipticity is exponentially integrable, thus not necessarily bounded. The right spaces of the gradient of the solutions to such equations turn out to be the Orlicz-Zygmund spaces $L^p\log^\alpha L(\Omega)$. It is the first time the regularity results for genuine nonisotropic PDEs have been successfully treated. Recent advances in harmonic analysis, especially the ${\mathcal H }^{1}-BMO$ duality theory, play the key role in our arguments. Applications to the geometric function theory are already in place [16].

This paper deals with the existence of positive solutions of problem $-\Delta u=u^{N+2\over N-2}+{\varepsilon} w(x)u^q $, with Dirichlet zero boundary condition on $\Omega$ (a bounded domain in $\mathbb R^N$), when $q\geq 1$ and $q\neq{N+2\over N-2}$. We study the existence of solutions which blow-up and concentrate at a single point of $\Omega$ whose location depends on the Robin function and on the coefficient $w$ of the perturbed term.

We consider positive solutions of the semilinear heat equation $$u_t=\Delta u+u^p,\quad \hbox{ in $(0,T)\times {\mathbb R}^N$,} \tag*{(1)} $$ with $p>1$ if $N=1$ or $2$ and $1 < p < {N+2\over N-2}$ if $N=3$. We show that the blow-up rate of all radially decreasing solutions of (1) satisfies a universal global a priori estimate. Namely, for all $\varepsilon\in (0,1)$, we prove that $$ \|u(t)\|_\infty\leq C(T-t)^{-1/(p-1)},\quad \varepsilon T < t < T, \tag*{(2)} $$ where $C=C(N, p, \varepsilon)>0$ is {\it independent of} $u$. This estimate has various applications. In particular, it implies a strong uniform decay property for global solutions of (1), which seems to have been conjectured in previous works on equation (1). Namely, all global positive radially decreasing solutions of (1) decay at least like $t^{-1/(p-1)}$. Also, as consequences of (2), we derive a parabolic Liouville theorem for (1), and for some equations of the form $u_t=\Delta u+f(u,\nabla u)$, we obtain results on blow-up rates and a priori estimates of global solutions. The proof of (2) relies on smoothing estimates in the uniformly local Lebesgue spaces $L^q_{\rho,\star}$, obtained in a previous work of the authors, together with the use of self-similar variables and suitable energy arguments.