Global multiplicity for very-singular elliptic problems with vanishing non-local terms

Abstract

In this paper, we deal with issues related to global multiplicity of $W^{1,p}_{\mathrm {loc}}(\Omega )$-solutions for the very-singular and non-local $\mu$-problem

$$\begin{aligned} -{g\left( \int \limits _\Omega u^q\right) }\Delta _pu={\mu } u^{-\delta } + u^{\beta } \ \ \text{ in } \ \ \Omega , \ \ \ \ u > 0 \ \ \ \text{ in } \ \Omega \ \ \ \text{ and } \ \ u=0 \ \ \text{ on } \ \partial \Omega , \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ is a smooth bounded domain, $\delta >0$, $q>0$, $0 < \beta \le p-1$ and $g{:}\,[0,\infty ) \rightarrow [0,\infty )$ is a continuous function that achieves critical values for the class of non-local problems (i.e., the level zero if $\beta < p-1$ and $1/\lambda _1$ if $\beta = p-1$, where $\lambda _1$ stands for the principal eigenvalue of the p-Laplacian in $\Omega$ under homogeneous Dirichlet boundary conditions). To overcome the difficulties arising from the geometry of g and the presence of very-singular term combined with a $(p-1)$-sublinear/asymptotically linear ones, we take advantage of a comparison principle for sub-supersolutions in $W^{1,p}_{\mathrm {loc}}(\Omega )$-sense proved in Santos and Santos (Z Angew Math Phys 69:Art. 145, 2018), together with sub-supersolutions techniques and bifurcation theory.