Continuums of positive solutions for classes of non-autonomous and non-local problems with strong singular term

Abstract

In this paper, we show existence of \textit{continuums} of positive solutions for non-local quasilinear problems with strongly-singular reaction term on a bounded domain in ${\mathbb{R}}^N$ with $N\ge 2$. We approached non-autonomous and non-local equations by applying the Bifurcation Theory to the corresponding $\epsilon$-perturbed problems and using a comparison principle for $W_{\mathrm{loc}}^{1,p}(\Omega )$-sub and supersolutions to obtain qualitative properties of the $\epsilon$-\textit{continuum} limit. Moreover, this technique empowers us to study a strongly-singular and non-homogeneous Kirchhoff problem to get the existence of a \textit{continuum} of positive solutions.

24 pages, 4 figures