The singularity problems in nonlinear elliptic equations: history and progress. Lecture 2

We give an overview of the old and more recent developments of the study of the singularity problem for quasilinear elliptic equations in a domain of $\mathbb{R}^N$
$$ -div A(x,u,\nabla u)+B(x,u,\nabla u)=0 $$
since the pioneering works of James Serrin (1964-1965). The problem is twofold:
1- If the above equation is satisfied in a punctured domain say $B_1\setminus\{0\}$, is it possible to describe the behaviour of $u(x)$ when $x\to 0$ ?
2- If the above equation is satisfied in $B_1\setminus \Sigma$ where $\Sigma$ is a subset of $B_1$, under what conditions the function can be extended as a solution of the same equation in whole $\Omega$ (we say that $\Sigma$ is a removable singularity) ?
Examples are
$$A(x,u,\nabla u)=|\nabla u|^{p-2}\nabla u$$
and
$$B(x,u,\nabla u)=\pm |u|^{q-1}u\;±, \;B(x,u,\nabla u)= \pm |\nabla u|^r\quad \text{or }\;B(x,u,\nabla u)= |u|^{q-1}u\pm |\nabla u|^r.$$
We will recall that Serrin's assumptions are (with $1<m\leq N$)
$$A(x,u,\nabla u)\sim |\nabla u|^{m-2}\nabla u\, \text{ and }\;|B(x,u,\nabla u)|\leq c(|u|^{m-1}+ |\nabla u|^{m-1}), $$
and in his case the pertubation term $B$ plays a minor role. This is the contrary in the two fundamental superlinear cases that we will present: Lane-Emden's equation $ -\Delta u-u^q=0 $ and Emden-Fowler's equations $ -\Delta u+u^q=0 $ where $q>1$ and $u\geq 0$.