Corrigendum and addendum to ``Non-autonomous quasilinear elliptic equations and Ważewski's principle''

Abstract

In this addendum we fill a gap in a proof and we correct some results appearing in [12]. In the original paper [12] we classified positive solutions for the following equation \begin{equation*} \Delta_{p}u+K(r) u^{\sigma-1}=0 \end{equation*} where $r=|x|$, $x \in \mathbb R^n$, $n >p>1$, $\sigma ={n p}/({n-p})$ and $K(r)$ is a function strictly positive and bounded. In fact [12] had two main purposes. First, to establish asymptotic conditions which are sufficient for the existence of ground states with fast decay and to classify regular and singular solutions: these results are correct but need some non-trivial further explanations. Second to establish some computable conditions on $K$ which are sufficient to obtain multiplicity of ground states with fast decay in a non-perturbation context. Also in this case the original argument contained a flaw: here we correct the assumptions of [12] by performing a new nontrivial construction. A third purpose of this addendum is to generalize results of [12] to a slightly more general equation \begin{equation*} \Delta_p u+ r^{\delta}K(r) u^{\sigma(\delta)-1}=0 \end{equation*} where $\delta>-p$, and $\sigma(\delta) ={p(n+\delta)}/({n-p})$.