Singularities for nonlinear elliptic equations with singular potentials and gradient-dependent lower-order terms. Lecture 1

Lecture 1: We discuss various methods to obtain sharp existence and classification results for the positive solutions of nonlinear elliptic equations in $\mathbb R^N\setminus \{0\}$. Problems of this type have been studied extensively by many authors. For the prototype model $\Delta u=u^q$ in $\mathbb R^N\setminus \{0\}$ with $N\geq 3$, the pioneering paper of Brezis–Véron (1980/1981) shows that there are no positive solutions when $q\geq N/(N-2)$. When $1<q<N/(N-2)$, then the existence and profile near zero of all positive $C^1(\mathbb R^N\setminus\{0\})$-solutions are given by Friedman and Véron (1986). We provide the counterpart of these results for elliptic equations featuring a Hardy potential and weighted nonlinearities. Lecture 1 is based on joint work with M. Fărcăşeanu [J. Differential Equations 292 (2021), 461–500].
Lecture 2: We consider the positive solutions to the model problem $\Delta u=u^q |\nabla u|^m$ in $\Omega\setminus \{0\}$, where $\Omega$ is a domain in $\mathbb R^N$ $(N\geq 2)$ with $0\in \Omega$. We assume that $q\geq 0$, $m\in (0,2)$ and $m+q>1$. We present methods to obtain a complete classification of the behaviour near zero, as well as at $\infty$ when $\Omega=\mathbb R^N$, of the positive solutions of the above problem, together with corresponding existence results. This study is motivated by a rich literature on the topic of isolated singularities (e,g, Serrin (1965), Brezis–Oswald (1987), Vázquez–Véron (1980; 1985); Véron (1981; 1986; 1996), Nguyen Phuoc–Véron (2012) and Marcus–Nguyen (2015)). We emphasise the changes that arise from the introduction of the gradient factor in the nonlinear term and the new phenomena emerging when $m\in (0,1)$. Lecture 2 is based on joint work with J. Ching [Anal. PDE, 8 (8) (2015), 1931–1962].