| Abstract: |
| Here we consider the nonnegative solutions of equations in a punctured ball
$B(0,R)\setminus\left\{ 0\right\} \subset\mathbb{R}^{N}$ or in
$\mathbb{R}^{N},$ of type
\begin{equation}
-\Delta u=u^{p}+M|\nabla u|^{q}\label{two}%
\end{equation}
where $p,q>1$ and $M\in\mathbb{R}$. We give new a priori estimates on the solutions and their gradient, and Liouville type results. We use Bernstein technique and Osserman`s or Gidas-Spruck`s type methods. The most
interesting case is $q=2p/(p+1)$, where the equation is invariant by scaling.
In the radial case, we give a precise description of all the solutions,
improving the known results. The situation appears to be quite complicated in
the case $M |
|