How to break the uniqueness of $W_{loc}^{1,p}(\Omega )$-solutions for very singular elliptic problems by non-local terms

Abstract

In this paper, we are going to show existence of branches of bifurcation for positive $W_{loc}^{1,p}(\Omega )$-solutions for the very singular non-local $\lambda$-problem $$ -{\Big(\int_Ωg(x,u)dx\Big)^r}Δ_pu={λ} \Big(a(x)u^{-δ} + b(x)u^β\Big) \ \ \mbox{in} \ \ Ω, \ \ \ \ u > 0 \ \ \ \mbox{in} \ Ω\ \ \ \mbox{and} \ \ u=0 \ \ \mbox{on} \ \partial Ω, $$ where $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $\delta >0$, $0<\beta <p-1$, $a$ and $b$ are non-negative measurable functions and $g$ is a positive continuous function. Our approach is based on sub-supersolutions techniques, fixed point theory, in the study of $W_{loc}^{1,p}(\Omega )$-topology of a solution application and a new comparison principle for sub-supersolutions in $W_{loc}^{1,p}(\Omega )$ to a problem with $p$-Laplacian operator perturbed by a very singular term at zero and sublinear at infinity.