How to break the uniqueness of $W_{loc}^{1,p}(\mathrm{\Omega })$-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 $W_{loc}^{1,p}(\mathrm{\Omega })$-solutions for the very singular non-local $\lambda$-problem $$ -{\Big(\int_\Omega g(x,u)dx\Big)^r}\Delta_pu={\lambda } \Big(a(x)u^{-\delta}

• b(x)u^{\beta}\Big) \ \ \mbox{in} \ \ \Omega, \ \ \ \ u > 0 \ \ \ \mbox{in}
  \Omega \ \ \ \mbox{and} \ \ u=0 \ \ \mbox{on} \ \partial \Omega, $$where$\Omega \subset $\mathbb{R}$^N $isasmoothboundeddomain,$\delta >0$,$0 < \beta < p-1$,$a $and$b$arenon-negativemeasurablefunctionsand$g$ is a positive continuous function. Our approach is based on sub-supersolutions techniques, fixed point theory, in the study of $W_{loc}^{1,p}(\mathrm{\Omega })$-topology of a solution application and a new comparison principle for sub-supersolutions in $W_{loc}^{1,p}(\mathrm{\Omega })$ to a problem with $p$-Laplacian operator perturbed by a very singular term at zero and sublinear at infinity.