How to break the uniqueness of $W^{1,p}_{loc}(Ω)$-solutions for very singular elliptic problems by non-local terms
In this paper, we are going to show existence of branches of bifurcation for positive $W^{1,p}_{loc}(Ω)$-solutions for the very singular non-local $λ$-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 $Ω\subset \mathbb{R}^N $ is a smooth bounded domain, $δ>0$, $0 < β< p-1$, $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^{1,p}_{loc}(Ω)$-topology of a solution application and a new comparison principle for sub-supersolutions in $W^{1,p}_{loc}(Ω)$ to a problem with $p$-Laplacian operator perturbed by a very singular term at zero and sublinear at infinity.