Strongly singular problems involving the (1, q)-Laplacian

Abstract In this paper, we study a strongly singular Dirichlet problem driven by the (1, q )-Laplacian of the form $$\begin{aligned} -\operatorname {div} \left( \frac{Du}{|Du|}\right) -\Delta _qu = \frac{f(x)}{{\left( u^+\right) }^\gamma } \quad \text {in } \Omega ,\quad u = 0 \quad \text {on } \partial \Omega , \end{aligned}$$ - div Du | D u | - Δ q u = f ( x ) u + γ in Ω , u = 0 on ∂ Ω , where $$1< q < N$$ 1 < q < N , $$\gamma > 1$$ γ > 1 , and $$f\in L^N(\Omega )$$ f ∈ L N ( Ω ) with $$f(x)>0$$ f ( x ) > 0 for a.e. $$x\in \Omega $$ x ∈ Ω , where $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N , $$N\ge 2$$ N ≥ 2 , is a bounded domain with Lipschitz boundary $$\partial \Omega $$ ∂ Ω . The interplay between the degenerate 1-Laplacian a