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

Matteo Franca

DOI: http://dx.doi.org/10.12775/TMNA.2019.110

Abstract


In this addendum we fill a gap in a proof and we correct some results appearing in \cite{F6}. In the original paper \cite{F6} we classified positive solutions for the following equation \begin{equation*} \Delta_{p}u+K(r) u^{\sigma-1}=0 \end{equation*} where $r=|x|$, $x \in \RR^n$, $n> p> 1$, $\sigma ={n p}/({n-p})$ and $K(r)$ is a function strictly positive and bounded. In fact \cite{F6} 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 \cite{F6} by performing a new nontrivial construction. A third purpose of this addendum is to generalize results of \cite{F6} 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 $\sigma(\delta) ={p(n+\delta)}/({n-p})$.

Keywords


$p$-laplace equations; invariant manifold; non-smooth systems; radial solutions; ground states; Fowler transformation; Ważewski's principle

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