How to break the uniqueness of $W^{1,p}_{loc}(\Omega)$-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 $W^{1,p}_{loc}(\Omega)$-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 $ 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^{1,p}_{loc}(\Omega)$-topology of a solution application and a new comparison principle for sub-supersolutions in $W^{1,p}_{loc}(\Omega)$ to a problem with $p$-Laplacian operator perturbed by a very singular term at zero and sublinear at infinity.