A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems

In this paper we deal with the elliptic problem $$\begin{array}{} \begin{cases} \displaystyle-{\it\Delta} u=\lambda u+\mu(x)\frac{|\nabla u|^q}{u^\alpha}+f(x)\quad &\text{ in }{\it\Omega}, \\ u \gt 0 \quad &\text{ in }{\it\Omega}, \\ u=0\quad &\text{ on }\partial{\it\Omega}, \end{cases} \end{array} $$ where Ω ⊂ ℝN is a bounded smooth domain, 0 ≨ μ ∈ L∞(Ω), 0 ≨ f ∈ Lp0(Ω) for some p0 > $\begin{array}{} \frac{N}{2} \end{array}$, 1 < q < 2, α ∈ [0 1] and λ ∈ ℝ. We establish existence and multiplicity results for λ > 0 and α < q – 1, including the non-singular case α = 0. In contrast, we also derive existence and uniqueness results for λ > 0 and q – 1 < α ≤ 1. We thus complement the results in [1, 2], which are concerned with α = q – 1, and show that the value α = q – 1 plays the role of a break point for the multiplicity/uniqueness of solution.